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A 


FIRST  BOOK  IN  ALGEBRA 


BY 


WALLACE  C.  BOY  DEN,  A.M. 

8i7b-Masteb  of  the  Boston  Normal  School 


SILVER,   BURDETT   AND  COMPANY 

New  York         BOSTON  Chicago 

1894 


i^ 


T 


Copyright,  1894, 
By  silver,  BURDETT  &  COMPANY. 


Nortoooti  ^ress : 

J.  S  Gushing  &  Co.  —  Berwick  &  Smith. 

Boston,  Mass.,  U.S.A. 


PREFACE. 


In  preparing  this  book,  the  author  had  especially  in 
mind  classes  in  the  upper  grades  of  grammar  schools, 
though  the  work  will  be  found  equally  well  adapted  to 
the  needs  of  any  classes  of  beginners. 

The  ideas  which  have  guided  in  the  treatment  of  the 
subject  are  the  following :  The  study  of  algebra  is  a 
continuation  of  what  the  pupil  has  been  doing  for  years, 
but  it  is  expected  that  this  new  work  will  result  in  a 
knowledge  of  general  truths  about  numbers,  and  an  in- 
creased power  of  clear  thinking.  All  the  differences 
between  this  work  and  that  pursued  in  arithmetic  may 
be  traced  to  the  introduction  of  two  new  elements,  namely, 
negative  numbers  and  the  representation  of  numbers  by 
letters.  The  solution  of  problems  is  one  of  the  most 
valuable  portions  of  the  work,  in  that  it  serves  to  develop 
the  thought-power  of  the  pupil  at  the  same  time  that  it 
broadens  his  knowledge  of  numbers  and  their  relations. 
Powers  are  developed  and  habits  formed  only  by  per- 
sistent, long-continued  practice. 

Accordingly,  in  this  book,  it  is  taken  for  granted  that 
the  pupil  knows  what  he  may  be  reasonably  expected  to 
have    learned   from   his    study   of    arithmetic;    abundant 

M  1611 


4  PREFACE. 

practice  is  given  in  the  representation  of  numbers  by 
letters,  and  great  care  is  taken  to  make  clear  the  mean- 
ing of  the  minus  sign  as  applied  to  a  single  number, 
together  with  the  modes  of  operating  upon  negative  num- 
bers ;  problems  are  given  in  every  exercise  in  the  book ; 
and,  instead  of  making  a  statement  of  what  the  child  is 
to  see  in  the  illustrative  example,  questions  are  asked 
which  shall  lead  him  to  find  for  himself  that  which  he 
is  to  learn  from  the  example. 

Boston,  Mass.,  December,  1893. 


CONTENTS. 


PAGB 

ALGEBRAIC  NOTATION 7 

Problbms 7 

Modes  of  Representing  the  Operations 30 

Addition 30 

Subtraction 32 

Multiplication 30 

Division 38 

Algebraic  Expressions 39 

OPERATIONS 43 

Addition 43 

Subtraction 46 

Parentheses 50 

Multiplication 53 

Involution 68 

Division 62 

Evolution 69 

FACTORS  AND  MULTIPLES 76 

Factoring  —  Six  Cases 76 

Greathst  Common  Factor 88 

Least  Common  Multiple 90 

6 


6  CONTENTS. 

PAGE 

FRACTIONS 94 

Reduction  of  Fractions ,  94 

Operations  upon  Fractions 99 

Addition  and  Subtraction 99 

Multiplication  and  Division 100 

Involution,  Evolution,  and  Factoring , ...  112 

Complex  Fractions 114 

EQUATIONS 119 

Simple. 119 

Simultaneous 137 

Quadratic 141 


A  FIRST  BOOK  IN  ALGEBRA. 


»>0?c 


ALGEBRAIC   NOTATION/ 

1.  Algebra  is  so  much  like  arithmetic  that  all  that 
you  know  about  addition,  subtraction,  multiplication, 
and  division,  the  signs  that  you  have  been  using  and 
the  ways  of  working  out  problems,  will  be  very  useful 
to  you  in  this  study.  There  are  two  things  the  intro- 
duction of  which  really  makes  all  the  difference  be- 
tween arithmetic  and  algebra.  One  of  these  is  the 
use  of  letters  to  represent  numbers^  and  you  will  see 
in  the  following  exercises  that  this  change  makes  the 
solution  of  problems  much  easier. 

Exercise  1. 

Illustrative  Example.  The  sum  of  two  numbers  is  60,  and 
the  greater  is  four  times  the  less.     What  are  the  numbc^rs  ? 

Solution. 

Let  X  =  the  less  number  ; 

then  4x  =  the  greater  number, 

and  4x-\-  x  =  60, 

or  6  X  =  60  ; 

therefore  a;  =  12, 

and  4a;  =  48.          The  numbers  are  12  and  48. 


8  A    FIRST  BOOK   IN  ALGEBRA. 

1.  The  greater  of  two  numbers  is  twice  the  less,  and  the 
sum  of  the  numbers  is  129.     What  are  the  numbers  ? 

2.  A  man  bought  a  horse  and  carriage  for  $500,  paying 
three  times  as  much  for  the  carriage  as  for  the  horse.  How 
much  did  each  .cost  ? 

;  3.    Two;  bi'^t'luers,  counting  their  money,  found  that  to- 
,   .  .^  ^(^ether,  they  ^a.d  $1-86,  and  that  John  had  five  times  as 
' •  •'  nmehas  Ghal'les.'    How  much  had  each  ? 

4.  Divide  the  number  64  into  two  parts  so  that  one  part 
shall  be  seven  times  the  other. 

5.  A  man  walked  24  miles  in  a  day.  If  he  walked  twice 
as  far  in  the  forenoon  as  in  the  afternoon,  how  far  did  he 
walk  in  the  afternoon  ? 

6.  For  72  cents  Martha  bought  some  needles  and  thread, 
paying  eight  times  as  much  for  the  thread  as  for  the 
needles.     How  much  did  she  pay  for  each  ? 

7.  In  a  school  there  are  672  pupils.  If  there  are  twice 
as  many  boys  as  girls,  how  many  boys  are  there  ? 

Illustrative  Example.  If  the  difference  between  two 
numbers  is  48,  and  one  number  is  five  times  the  other, 
what  are  the  numbers  ? 


Solution. 

Let 

X  =  the  less  number  ; 

then 

6x  =  the  greater  number, 

and 

lyx  -  x  =  48, 

or 

4a;  =  48; 

therefore 

JK=12, 

and 

5a:  =  60. 

The  numbers  are  12  and  60. 


PHuULEMS.  9 

8.  Find  two  numbers  such  that  their  difference  is  250 
and  one  is  eleven  times  the  other. 

9.  James  gathered  12  quarts  of  nuts  more  than  Henry 
gathered.  How  many  did  each  gather  if  James  gathered 
tliree  times  as  many  as  Henry  ? 

10.  A  house  cost  $2880  more  than  a  lot  of  land,  and 
live  times  the  cost  of  the  lot  equals  the  cost  of  the  house. 
What  was  the  cost  of  each  ? 

11.  Mr.  A.  is  48  years  older  than  his  son,  but  he  is  only 
three  times  as  old.     How  old  is  each  ? 

12.  Two  farms  differ  by  250  acres,  and  one  is  six  times 
as  large  as  the  other.     How  many  acres  in  each  ? 

13.  William  paid  eight  times  as  much  for  a  dictionary 
as  for  a  rhetoric.  If  the  difference  in  price  was  $6.30, 
how  much  did  he  pay  for  each  ? 

14.  The  sum  of  two  numbers  is  4256,  and  one  is  37 
times  as  great  as  the  other.     What  are  the  numbers  ? 

15.  Aleck  has  48  cents  more  than  Arthur,  and  seven 
times  Arthur's  money  equals  Aleck's.  How  much  has 
each  ? 

16.  The  sum  of  the  ages  of  a  mother  and  daughter  is 
32  years,  and  the  age  of  the  mother  is  seven  times  that  of 
the  daughter.     What  is  the  age  of  each  ? 

17.  John's  age  is  three  times  that  of  Mary,  and  he  is 
10  years  older.     What  is  the  age  of  each  ? 


10  A    FIRST  BOOK   IN  ALGEBRA. 

Exercise  2. 

lUustrative  Example.  There  are  three  numbers  whose 
sum  is  96 ;  the  second  is  three  times  the  first,  and  the  third 
is  four  times  the  first.     What  are  the  numbers  ? 

Solution. 

Let  X  =  first  number, 

Sx  =  second  number, 
4x  =  third  number. 
x  +  Sx-h  ^x  =  9Q 
8x  =  96 

X=:  12 

3a;  =  SQ 

The  numbers  are  12,  36,  and  48. 

1.  A  man  bought  a  hat,  a  pair  of  boots,  and  a  necktie 
for  $7.50;  the  hat  cost  four  times  as  much  as  the  necktie, 
and  the  boots  cost  five  times  as  much  as  the  necktie.  What 
was  the  cost  of  each  ? 

2.  A  man  traveled  90  miles  in  three  days.  If  he  trav- 
eled twice  as  far  the  first  day  as  he  did  the  third,  and 
three  times  as  far  the  second  day  as  the  third,  how  far 
did  he  go  each  day  ? 

3.  James  had  30  marbles.  He  gave  a  certain  number 
to  his  sister,  twice  as  many  to  his  brother,  and  had  three 
times  as  many  left  as  he  gave  his  sister.  How  many  did 
each  then  have  ? 


PROBLEMS.  ll 

4.  A  farmer  bought  a  horse,  cow,  and  pig  for  $90.  If 
he  paid  three  times  as  much  for  the  cow  as  for  the  pig,  and 
iive  times  as  much  for  the  horse  as  for  the  pig,  what  was 
the  price  of  each  ? 

5»  A  had  seven  times  as  many  apples,  and  B  three  times 
as  many  as  C  had.  If  they  all  together  had  55  apples,  how 
many  had  each  ? 

6.  The  difference  between  two  numbers  is  36,  and  one 
is  four  times  the  other.     What  are  the  numbers? 

7.  In  a  company  of  48  people  there  is  one  man  to  each 
five  women.     How  many  are  there  of  each  ? 

8.  A  man  left  $1400  to  be  distributed  among  three 
sons  in  such  a  way  that  James  was  to  receive  double 
what  John  received,  and  John  double  what  Henry  received. 
How  much  did  each  receive  ? 

9.  A  field  containing  45,000  feet  was  divided  into  three 
lots  so  that  the  second  lot  was  three  times  the  first,  and 
the  third  twice  the  second.     How  large  was  each  lot? 

10.  There  are  120  pigeons  in  three  flocks.  In  the  second 
there  are  three  times  as  many  as  in  the  first,  and  in  the 
third  as  many  as  in  the  first  and  second  combined.  How 
many  pigeons  in  each  flock? 

11.  Divide  209  into  three  parts  so  that  the  first  part 
shall  be  five  times  the  second,  and  the  second  three  times 
the  third. 

12.  Three  men.  A,  B,  and  C,  earned  $110;  A  earned 
four  times  as  much  as  B,  and  C  as  much  as  both  A  and  B. 
How  much  did  each  earn  ? 


12  A    FIRST  BOOK  IJ^  ALGEBRA. 

13.  A  faTmer  bought  a  horse,  a  cow,  and  a  calf  for  $72; 
the  cow  cost  twice  as  much  as  the  calf,  and  the  horse  three 
times  as  much  as  the  cow.     What  was  the  cost  of  each  ? 

14.  A  cistern,  containing  1200  gallons  of  water,  is  emp- 
tied by  two  pipes  in  two  hours.  One  pipe  discharges  three 
times  as  many  gallons  per  hour  as  the  other.  How  many 
gallons  does  each  pipe  discharge  in  an  hour? 

15.  A  butcher  bought  a  cow  and  a  lamb,  paying  six  times 
as  much  for  the  cow  as  for  the  lamb,  and  the  difference  of 
the  prices  was  $  25.     How  much  did  he  pay  for  each  ? 

16.  A  grocer  sold  one  pound  of  tea  and  two  pounds  of 
coffee  for  f  1.50,  and  the  price  of  the  tea  per  pound  was 
three  times  that  of  the  coffee.    What  was  the  price  of  each  ? 

17.  By  will  Mrs.  Cabot  was  to  receive  five  times  as 
much  as  her  son  Henry.  If  Henry  received  $20,000  less 
than  his  mother,  how  much  did  each  receive  ? 

Exercise  3.. 

Illustrative  Examj^le.  Divide  the  number  126  into  two 
parts  such  that  one  part  is  8  more  than  the  other. 

Solution. 

Let  X  =  less  part, 

X  -\-  S  =  greater  part, 
x  +  x  +  8  =:  126 
2x  +  8  =  126 
2x  =  118* 
a;  =  59 
X  +  8  =  67 
The  parts  are  59  and  67, 

*  Where  in  arithmetic  did  you  lea»n  the  principle  applied  in  transposing  the  8  ? 


PROBLEMS.  13 

1.  In  a  class  of  35  pupils  there  are  7  more  girls  than 
boys.     How  many  are  there  of  each? 

2.  The  sum  of  the  ages  of  two  brothers  is  43  years,  and 
one  of  them  is  15  years  older  than  the  other.     Find  their 

ages. 

3.  At  an  election  in  which  1079  votes  were  cast  the  suc- 
cessful candidate  had  a  majority  of  95.  How  many  votes 
did  each  of  the  two  candidates  receive  ? 

4.  Divide  the  number  70  into  two  parts,  such  that  one 
part  shall  be  26  less  than  the  other  part. 

5.  John  and  Henry  together  have  143  marbles.  If  I 
should  give  Henry  15  more,  he  would  have  just  as  many 
as  John.    How  many  has  each  ? 

6.  In  a  storehouse  containing  57  barrels  there  are  3 
less  barrels  of  flour  than  of  meal.     How  many  of  each? 

7.  A  man  whose  herd  of  cows  numbered  63  had 
17  mere  Jerseys  than  Holsteins.  How  many  had  he  of 
each  ? 

8.  Two  men  whose  wages  differ  by  8  dollars  receive 
both  together  $44  per  month.  How  much  does  each 
receive  ? 

9.  Find  two  numbers  whose  sum  is  99  and  whose  dif- 
ference is  19. 

10.  The  sum  of  three  numbers  is  56;  the  second  is  3 
more  than  the  first,  and  the  third  5  more  than  the  first. 
What  are  the  numbers? 


14  A    FIRST  BOOK  IN  ALGEBRA. 

11.  Divide  62  into  three  parts  such  that  the  first  part 
is  4  more  than  the  second,  and  the  third  7  more  than  the 
second. 

12.  Three  men  together  received  $34,200;  if  the  second 
received  $1500  more  than  the  first,  and  the  third  $1200 
more  than  the  second,  how  much  did  each  receive  ? 

13.  Divide  65  into  three  parts  such  that  the  second  part 
is  17  more  than  the  first  part,  and  the  third  15  less  than  the 
first. 

14.  A  man  had  95  sheep  in  three  flocks.  In  the  first 
flock  there  were  23  more  than  in  the  second,  and  in  the 
third  flock  12  less  than  in  the  second.  How  many  sheep 
in  each  flock  ? 

15.  In  an  election,  in  which  1073  ballots  were  cast,  Mr. 
A  receives  97  votes  less  than  Mr.  B,  and  Mr.  C  120  votes 
more  than  Mr.  B.     How  many  votes  did  each  receive  ? 

16.  A  man  owns  three  farms.  In  the  first  there  are 
5  acres  more  than  in  the  second  and  7  acres  less  than  in 
the  third.  If  there  are  53  acres  in  all  the  farms  together, 
how  many  acres  are  there  in  each  farm  ? 

17.  Divide  111  into  three  parts  so  that  the  first  part 
shall  be  16  more  than  the  second  and  19  less  than  the 
third. 

18.  Three  firms  lost  $118,000  by  fire.  The  second  firm 
lost  $  6000  less  than  the  first  and  $  20,000  more  than  the 
third.     What  was  each  firm's  loss  ? 


PROBLEMS.  15 

Exercise  4. 

Illustrative  Example.  The  sum  of  two  numbers  is  25,  and 
tlie  larger  is  3  less  than  three  times  the  smaller.  What 
;ire  the  numbers  ? 

•  Solution. 

Let  X  =  smaller  number, 

3aj  —  3  =  larger  number, 
x-f  3x-3  =  25 
4a;-3  =  26v 
4a;  =  28* 
z  =  7 
3x-3  =  18 
The  numbers  are  7  and  18. 

1.  Charles  and  Henry  together  have  49  marbles,  and 
Charles  has  twice  as  many  as  Henry  and  4  more.  How 
many  marbles  has  each  ? 

2.  In  an  orchard  containing  33  trees  the  number  of 
pear  trees  is  5  more  than  three  times  the  number  of  apple 
trees.     How  many  are  there  of  each  kind? 

3.  John  and  Mary  gathered  23  quarts  of  nuts.  John 
gathered  2  quarts  more  than  twice  as  many  as  Mary.  How 
many  quarts  did  each  gather  ? 

4.  To  the  double  of  a  number  I  add  17  and  obtain  as 
a  result  147.     What  is  the  number  ? 

5.  To  four  times  a  number  I  add  23  and  obtain  95. 
What  is  the  number? 

6.  From  three  times  a  number  I  take  25  and  obtain  47. 
What  is  the  number  ? 

*  Ia  the  same  principle  applied  here  that  is  applied  on  page  12  ? 


16  A    FIRST  BOOK  IN  ALGEBRA. 

7.  Find  a  number  which  being  multiplied  by  5  and 
having  14  added  to  the  product  will  equal  69. 

8.  I  bought  some  tea  and  coffee  for  ^  10.39.  If  I  paid 
for  the  tea  61  cents  more  than  five  times  as  much  as  for 
the  coffee,  how  much  did  I  pay  for  each? 

9.  Two  houses  together  contain  48  rooms.  If  the 
second  house  has  3  more  than  twice  as  many  rooms  as  the 
first,  how  many  rooms  has  each  house  ? 

Illustrative  Exam2Jle.  Mr.  Y  gave  ^  6  to  his  three  boys. 
To  the  second  he  gave  25  cents  more  than  to  the  third,  and 
to  the  first  three  times  as  much  as  to  the  second.  How 
much  did  each  receive  ? 

Solution. 

Let  X  =  number  of  cents  third  boy  received, 

a;  +  25  =  number  of  cents  second  boy  received, 
3  a:  +  75  =  number  of  cents  first  boy  received. 
X  +  X  +  25  +  3  X  +  75  =  600 
5x+100  =  600 
5x  =  500 
X  =  100 
X  +  25  =  125 
3x  +  75  =  375 

1st  boy  received  $  3.75, 
2d  boy  received  f$  1.25, 
3d  boy  received  $  1.00. 

10.  Divide  the  number  23  into  three  parts,  such  that  the 
second  is  1  more  than  the  first,  and  the  third  is  twice  the 
second. 


PROBLEMS.  17 

11.  Divide  the  number  137  into  three  parts,  such  that 
the  second  shall  be  3  more  than  the  first,  and  the  third 
five  times  the  second. 

12.  Mr.  Ames  builds  three  houses.  The  first  cost  $  2000 
more  than  the  second,  and  the  third  twice  as  much  as  the 
first.  If  they  all  together  cost  $  18,000,  what  was  the  cost 
of  each  house  ? 

13.  An  artist,  who  had  painted  three  pictures,  charged 
$  18  more  for  the  second  than  the  first,  and  three  times  as 
much  for  the  third  as  the  second.  If  he  received  $  322  for 
the  three,  what  was  the  price  of  each  picture  ? 

14.  Three  men,  A,  B,  and  C,  invest  $47,000  in  business. 
B  puts  in  $  500  more  than  twice  as  much  as  A,  and  C  puts 
in  three  times  as  much  as  B.  How  many  dollars  does  each 
put  into  the  business? 

16.  In  three  lots  of  land  there  are  80,750  feet.  The 
second  lot  contains  250  feet  more  than  three  times  as 
much  as  the  first  lot,  and  the  third  lot  contains  twice  as 
much  as  the  second.     What  is  the  size  of  each  lot  ? 

16.  A  man  leaves  by  his  will  $225,000  to  be  divided 
as  follows:  his  son  to  receive  $10,000  less  than  twice  as 
much  as  the  daughter,  and  the  widow  four  times  as  much 
as  the  son.     What  was  the  share  of  each  ? 

17.  A  man  and  his  two  sons  picked  25  quarts  of  berries. 
The  older  son  picked  5  quarts  less  than  three  times  as 
many  as  the  younger  son,  and  the  father  picked  twice  as 
many  as  the  older  son.     How  many  quarts  did  each  pick  ? 


18  A   FlliSl'  BOOK  IN  ALGEBRA. 

18.  Three  brothers  have  574  stamps.  John  has  15  less 
than  Henry,  and  Thomas  has  4  more  than  John.  How- 
many  has  each  ? 

Exercise  5. 

Illustrative  Example.  Arthur  bought  some  apples  and 
twice  as  many  oranges  for  78  cents.  The  apples  cost  3 
cents  apiece,  and  the  oranges  5  cents  apiece.  How  many  of 
each  did  he  buy  ? 

Solution. 
Let  *      X  =  number  of  apples, 

2  X  =  number  of  oranges, 
3x  =  cost  of  apples, 
10  X  =  cost  of  oranges. 
3x+10x  =  78 
13x  =  78 
x  =  6 
2x=12 

Arthur  bought  6  apples  and  12  oranges. 

1.  Mary  bought  some  blue  ribbon  at  7  cents  a  yard,  and 
three  times  as  much  white  ribbon  at  5  cents  a  yard,  paying 
^  1.10  for  the  whole.  How  many  yards  of  each  kind  did 
she  buy  ? 

2.  Twice  a  certain  number  added  to  five  times  the 
double  of  that  number  gives  for  the  sum  36.  What  is  the 
number? 

3.  Mr.  James  Cobb  walked  a  certain  length  of  time  at 
the  rate  of  4  miles  an  hour,  and  then  rode  four  times  as 
long  at  the  rate  of  10  miles  an  hour,  to  finish  a  journey  of 
88  miles.     How  long  did  he  walk  and  how  long  did  he  ride  ? 


PROBLEMS.  19 

4.  A  man  bought  3  books  and  2  lamps  for  $14.  The 
price  of  a  lamp  was  twice  that  of  a  book.  What  was  the 
cost  of  each? 

5.  George  bought  an  equal  number  of  apples,  oranges, 
and  bananas  for  $  1.08 ;  each  apple  cost  2  cents,  each  orange 
4  cents,  and  each  banana  3  cents.  How  many  of  each  did 
he  buy  ? 

6.  I  bought  some  2-cent  stamps  and  twice  as  many 
5-cent  staipps,  paying  for  the  whole  $1.44.  How  many 
stamps  of  each  kind  did  I  buy  ? 

7.  I  bought  2  pounds  of  coffee  and  1  pound  of  tea  for 
$  1.31 ;  the  price  of  a  pound  of  tea  was  equal  to  that  of  2 
pounds  of  coffee  and  3  cents  more.  What  was  the  cost  of 
each  per  pound? 

8.  A  lady  bought  2  pounds  of  crackers  and  3  pounds  of 
gingersnaps  for  $1.11.  If  a  pound  of  gingersnaps  cost  7 
cents  more  than  a  pound  of  crackers,  what  was  tl^e  price  of 
each? 

9.  A  man  bought  3  lamps  and  2  vases  for  $6.  If  a 
vase  cost  50  cents  less  than  2  lamps,  what  was  the  price  of 
each? 

10.  I  sold  three  houses,  of  equal  value,  and  a  barn  for 
$1G,800.  If  the  barn  brought  $1200  less  than  a  house, 
what  was  the  price  of  each  ? 

11.  Five  lots,  two  of  one  size  and  three  of  another, 
aggregate  63,000  feet.  Each  of  the  two  is  1500  feet  larger 
than  each  of  the  three.     What  is  the  size  of  the  lots  ? 


20  A   FIRST  BOOK  IN  ALGEBRA. 

12.  Four  pumps,  two  of  one  size  and  two  of  another, 
can  pump  106  gallons  per  minute.  If  the  smaller  pumps 
5  gallons  less  per  minute  than  the  larger,  how  much  does 
each  pump  per  minute  ? 

13.  Johnson  and  May  enter  into  a  partnership  in  which 
Johnson's  interest  is  four  times  as  great  as  May's.  John- 
son's profit  was  $  4500  more  than  May's  profit.  What  was 
the  profit  of  each  ? 

14.  Three  electric  cars  are  carrying  79  persons.  In  the 
first  car  there  are  17  more  people  than  in  the  second 
and  15  less  than  in  the  third.  How  many  persons  in 
each  car  ? 

15.  Divide  71  into  three  parts  so  that  the  second  part 
shall  be  5  more  than  four  times  the  first  part,  and  the 
third  part  three  times  the  second. 

16.  I  bought  a  certain  number  of  barrels  of  apples  and 
three  times  as  many  boxes  of  oranges  for  $33.  I  paid 
$  2  a  barrel  for  the  apples,  and  $  3  a  box  for  the  oranges. 
How  many  of  each  did  I  buy  ? 

17.  Divide  the  number  288  into  three  parts,  so  that 
the  third  part  shall  be  twice  the  second,  and  the  second 
five  times  the  first. 

18.  Find  two  numbers  whose  sum  is  216  and  whose 
difference  is  48. 

Exercise  6. 
Illustrative    Example.      What   number   added   to    twice 
itself  and  40  more  will  make  a  sum  equal  to  eight  times 
the  number  ? 


PROBLEMS.  21 

Solution. 
Let  X  =  the  number. 

a;  +  2x  +  40  =  8a; 
3x  +  40  =  8a; 
40  =  6a; 
8  =  a; 
The  number  is  8. 

1.  What  number,  being  increased  by  36,  will  be  equal  to 
ten  times  itself  ? 

2.  Find  the  number  whose  double  increased  by  28  will 
equal  six  times  the  number  itself. 

3.  If  John's  age  be  multiplied  by  5,  and  if  24  be  added 
to  the  product,  the  sum  will  be  seven  times  his  age.  What 
is  his  age  ? 

4.  A  father  gave  his  son  four  times  as  many  dollars  as 
he  then  had,  and  his  mother  gave  him  $25,  when  he 
found  that  he  had  nine  times  as  many  dollars  as  at  first. 
How  many  dollars  had  he  at  first  ? 

5.  A  man  had  a  certain  amount  of  money ;  he  earned 
three  times  as  much  the  next  week  and  found  $32.  If 
he  then  had  eight  times  as  much  as  at  first,  how  much  had 
he  at  first  ? 

6.  A  man,  being  asked  how  many  sheep  he  had,  said, 
"  If  you  will  give  me  24  more  than  six  times  what  I  have 
now,  I  shall  have  ten  times  my  present  number."  How 
many  had  he  ? 

7.  Divide  the  number  726  into  two  parts  such  that  one 
shall  be  five  times  the  other. 


22  A    FIRST  BOOK   IN  ALGEBRA. 

8.  Find  two  numbers  differing  by  852,  one  of  which  is 
seven  times  the  other. 

9.  A  storekeeper  received  a  certain  amount  the  first 
month ;  the  second  month  he  received  $  50  less  than  three 
times  as  much,  and  the  third  month  twice  as  much  as  the 
second  month.  In  the  three  months  he  received  $4850. 
What  did  he  receive  each  month  ? 

10.  James  is  3  years  older  than  William,  and  twice 
James's  age  is  equal  to  three  times  William's  age.  What 
is  the  age  of  each  ? 

11.  One  boy  has  10  more  marbles  than  another  boy. 
Three  times  the  first  boy's  marbles  equals  five  times  the 
second  boy's  marbles.     How  many  has  each  ? 

12.  If  I  add  12  to  a  certain  number,  four  times  this 
second  number  will  equal  seven  times  the  original  number. 
What  is  the  original  number  ? 

13.  Four  dozen  oranges  cost  as  much  as  7  dozen  apples, 
and  a  dozen  oranges  cost  15  cents  more  than  a  dozen 
apples.     What  is  the  price  of  each  ? 

14.  Two  numbers  differ  by  6,  and  three  times  one 
number  equals  five  times  the  other  number.  What  are 
the  numbers  ? 

15.  A  man  is  2  years  older  than  his  wife,  and  15  times 
his  age  equals  16  times  her  age.  What  is  the  age  of 
each  ? 

16.  A  farmer  pays  just  as  much  for  4  horses  as  he  does 
for  6  cows.  If  a  cow  costs  15  dollars  less  than  a  horse, 
what  is  the  cost  of  each  ? 


PROBLEMS.  23 

17.  What  number  is  -that  which  is  15  less  than  four 
times  the  number  itself  ? 

18.  A  man  bought  12  pairs  of  boots  and  6  suits  of 
clothes  for  $168.  If  a  suit  of  clothes  cost  $2  less  than 
four  times  as  much  as  a  pair  of  boots,  what  was  the  price 
of  each  ? 

Exercise  7. 

Illustrative  Example.  Divide  the  number  72  into  two 
parts  such  that  one  part  shall  be  one-eighth  of  the  other. 

Solution. 

Let  X  =  greater  part, 

I  x  =  lesser  part. 

a;  +  J  x  =  72 
|a;  =  72 

x  =  Q4 
The  parts  are  64  and  8. 

1.  Roger  is  one-fourth  as  old  as  his  father,  and  the  sum 
of  their  ages  is  70  years.     How  old  is  each  ? 

2.  In  a  mixture  of  360  bushels  of  grain,  there  is  one- 
fifth  as  much  corn  as  wheat.     How  many  bushels  of  each  ? 

3.  A  man  bought  a  farm  and  buildings  for  $12,000. 
The  buildings  were  valued  at  one-third  as  much  as  the 
farm.     What  was  the  value  of  each  ? 

4.  A  bicyclist  rode  105  miles  in  a  day.  If  he  rode  one- 
half  as  far  in  the  afternoon  as  in  the  forenoon,  how  far  did 
he  ride  in  each  part  of  the  day  ? 


24  A   FIRST  BOOK  IN  ALGEBRA. 

5.  Two  numbers  diifer  by  675,  and  one  is  one-sixteenth 
of  the  other.     What  are  the  numbers  ? 

6.  What  number  is  that  which  being  diminished  by  one- 
seventh  of  itself  will  equal  162  ? 

7.  Jane  is  one-fifth  as  old  as  Mary,  and  the  difference 
of  their  ages  is  12  years.     How  old  is  each  ? 

Jllustrative  Example.  The  half  and  fourth  of  a  certain 
number  are  together  equal  to  75.     What  is  the  number  ? 

Solution. 
Let  X  =  the  number. 

ix+  Ja;  =  75 
I  a;  =  75 

a;  =  100 
The  number  is  100. 

8.  The  fourth  and  eighth  of  a  number  are  together 
equal  to  36.     What  is  the  number  ? 

9.  A  man  left  half  his  estate  to  his  widow,  and  a  fifth 
to  his  daughter.  If  they  both  together  received  $28,000, 
what  was  the  value  of  his  estate  ? 

10.  Henry  gave  a  third  of  his  marbles  to  one  boy,  and  a 
fourth  to  another  boy.  He  finds  that  he  gave  to  the  boys 
in  all  14  marbles.     How  many  had  he  at  first  ? 

11.  Two  men  own  a  third  and  two-fifths  of  a  mill  respec- 
tively. If  their  part  of  the  property  is  worth  $22,000, 
what  is  the  value  of  the  mill  ? 

12.  A  fruit-seller  sold  one-fourth  of  his  oranges  in  the 
forenoon,  and  three-fifths  of  them  in  the  afternoon.  If  he 
sold  in  all  255  oranges,  how  many  had  he  at  the  start  ? 


PROBLEMS,  26 

.13.  The  half,  third,  and  fifth  of  a  number  are  together 
equal  to  93.     Find  the  number. 

14.  Mr.  A  bought  one- fourth  of  an  estate,  Mr.  B  one-half, 
and  Mr.  C  one-sixth.  If  they  together  bought  55,000  feet, 
how  large  was  the  estate  ? 

15.  The  wind  broke  off  two-sevenths  of  a  pine  tree,  and 
afterwards  two-fifths  more.  If  the  parts  broken  off  meas- 
ured 48  feet,  how  high  was  the  tree  at  first  ? 

16.  A  man  spaded  up  three-eighths  of  his  garden,  and 
his  son  spaded  two-ninths  of  it.  In  all  they  spaded  43 
square  rods.     How  large  was  the  garden  ? 

17.  Mr.  A's  investment  in  business  is  $15,000  more  than 
Mr.  B's.  If  Mr.  A  invests  three  times  as  much  as  Mr.  B, 
how  much  is  each  man's  investment  ? 

18.  A  man  drew  out  of  the  bank  $27,  in  half-dollars, 
quarters,  dimes,  and  nickels,  of  each  the  same  number. 
What  was  the  number  ? 

Exercise  8. 

Illustrative  Example.  What  number  is  that  which  being 
increased  by  one-third  and  one-half  of  itself  equals  22  ? 

Solution. 

Let  X  =  the  number. 

l|a;  =  22 
ij^a;  =  22 
ix  =  2 
x=12 
The  number  is  12. 


26  A    FIRST  BOOK   IN  ALGEBRA. 

1.  Three  times  a  certain  number  increased  by  one-half 
of  the  number  is  equal  to  14.     What  is  the  number  ? 

2.  Three  boys  have  an  equal  number  of  marbles.  John 
buys  two-thirds  of  Henry's  and  two-fifths  of  Robert's 
marbles,  and  finds  that  he  then  has  93  marbles.  How 
many  had  he  at  first  ? 

3.  In  three  pastures  there  are  42  cows.  In  the  second 
there  are  twice  as  many  as  in  the  first,  and  in  the  third 
there  are  one-half  as  many  as  in  the  first.  How  many 
cows  are  there  in  each  pasture  ? 

4.  What  number  is  that  wliich  beiug  increased  by  one- 
half  and  one-fourth  of  itself,  and  5  more,  equals  33  ? 

5.  One-third  and  two-fifths  of  a  number,  and  11,  make 
44.     What  is  the  number  ? 

6.  What  number  increased  by  three-sevenths  of  itself 
will  amount  to  8640  ? 

7.  A  man  invested  a  certain  amount  in  business.  His 
gain  the  first  year  was  three-tenths  of  his  capital,  the 
second  year  five-sixths  of  his  original  capital,  and  the  third 
year  f  3600.  At  the  end  of  the  third  year  he  was  worth 
^  10,000.     What  was  his  original  investment  ? 

8.  Find  the  number  which,  being  increased  by  its  third, 
its  fourth,  and  34,  will  equal  three  times  the  number 
itself. 

9.  One-half  of  a  number,  two-sevenths  of  the  number, 
and  31,  added  to  the  number  itself,  will  equal  four  times 
the  number.     What  is  the  number  ? 


PROBLEMS.  27 

10.  A  man,  owning  a  lot  of  land,  bought  3  other  lots 
adjoining,  —  one  three-eighths,  another  one-third  as  large 
as  his  lot,  and  the  third  containing  14,000  feet,  —  when  he 
found  that  he  had  just  twice  as  much  land  as  at  first. 
How  large  was  his  original  lot  ? 

11.  What  number  is  doubled  by  adding  to  it  two-fifths 
of  itself,  one-third  of  itself,  and  8  ? 

12.  There  are  three  numbers  whose  sum  is  90 ;  the 
second  is  equal  to  one-half  of  the  first,  and  the  third  is 
equal  to  the  second  plus  three  times  the  first.  What  are 
the  numbers  ? 

13.  Divide  84  into  three  parts,  so  that  the  third  part 
shall  be  one-third  of  the  second,  and  the  first  part  equal  to 
twice  the  third  plus  twice  the  second  part. 

14.  Divide  112  into  four  parts,  so  that  the  second  part 
shall  be  one-fourth  of  the  first,  the  third  part  equal  to  twice 
the  second  plus  three  times  the  first,  and  the  fourth  part 
equal  to  the  second  plus  twice  the  first  part. 

15.  A  grocer  sold  62  i)ounds  of  tea,  coffee,  and  cocoa. 
Of  tea  he  sold  2  pounds  more  than  of  coffee,  and  of  cocoa 
4  pounds  more  than  of  tea.  How  many  pounds  of  each  did 
he  sell  ? 

16.  Three  houses  are  together  worth  six  times  as  much 
as  the  first  house,  the  second  is  worth  twice  as  much  as  the 
first,  and  the  third  is  worth  $7500.  How  much  is  each 
worth  ? 


28  A    FIRST  BOOK  IN  ALGEBRA. 

17.  John  has  one-ninth  as  much  money  as  Peter,  but  if 
his  father  should  give  him  72  cents,  he  would  have  just  the 
same  as  Peter.     How  much  money  has  each  boy  ? 

18.  Mr.  James  lost  two-fifteenths  of  his  property  in 
speculation,  and  three-eighths  by  fire.  If  his  loss  was 
$  6100,  what  was  his  property  worth  ? 

Exercise  9. 

1.  Divide  the  number  56  into  two  parts,  such  that  one 
part  is  three-fifths  of  the  other. 

2.  If  the  sum  of  two  numbers  is  42,  and  one  is  three- 
fourths  of  the  other,  what  are  the  numbers  ? 

3.  The  village  of  C is  situated  directly  between 

two  cities  72  miles  apart,  in  such  a  way  that  it  is  five- 
sevenths  as  far  from  one  city  as  from  the  other.  How  far 
is  it  from  each  city  ? 

4.  A  son  is  five-ninths  as  old  as  his  father.  If  the  sum 
of  their  ages  is  84  years,  how  old  is  each  ? 

5.  Two  boys  picked  26  boxes  of  strawberries.  If  John 
picked  five-eighths  as  many  as  Henry,  how  many  boxes  did 
each  pick  ? 

6.  A  man  received  60h  tons  of  coal  in  two  carloads,  one 
load  being  five-sixths  as  large  as  the  other.  How  many 
tons  in  each  carload  ? 

7.  John  is  seven-eighths  as  old  as  James,  and  the  sum 
of  their  ages  is  60  years.     How  old  is  each  ? 


PROBLEMS.  29 

8.  Two  men  invest  $1625  in  business,  one  putting  in 
five-eighths  as  much  as  the  other.  How  much  did  each 
invest  ? 

9.  In  a  school  containing  420  pupils,  there  are  three- 
fourths  as  many  boys  as  girls.  How  many  are  there  of 
each? 

10.  A  man  bought  a  lot  of  lemons  for  $5;  for  one-third 
lie  paid  4  cents  apiece,  and  for  the  rest  3  cents  apiece. 
How  many  lemons  did  he  buy  ? 

11.  A  lot  of  land  contains  15,000  feet  more  than  the 
adjacent  lot,  and  twice  the  first  lot  is  equal  to  seven  times 
the  second.     How  large  is  each  lot  ? 

12.  A  bicyclist,  in  going  a  journey  of  52  miles,  goes  a 
certain  distance  the  first  hour,  three-fifths  as  far  the  second 
hour,  one-half  as  far  the  third  hour,  and  10  miles  the  fourth 
hour,  thus  finishing  the  journey.  How  far  did  he  travel 
each  hour  ? 

18.  One  man  carried  off  three-sevenths  of  a  pile  of  loam, 
another  man  four-ninths  of  the  pile.  In  all  they  took  110 
cubic  yards  of  earth.     How  large  was  the  pile  at  first  ? 

14.  Matthew  had  three  times  as  many  stamps  as  Her- 
man, but  after  he  had  lost  70,  and  Herman  had  bought  90, 
they  put  what  they  had  together,  and  found  that  they  had 
540.     How  many  had  each  at  first  ? 

15.  It  is  required  to  divide  the  number  139  into  four 
parts,  such  that  the  first  may  be  2  less  than  the  second, 
7  more  than  the  third,  and  12  greater  than  the  fourth. 


30  A    FIRST  BOOK   IN  ALGEBRA. 

16.  In  an  election  7105  votes  were  cast  for  three  candi- 
dates. One  candidate  received  614  votes  less,  and  the  other 
1896  votes  less,  than  the  winning  candidate.  How  many- 
votes  did  each  receive  ? 

17.  There  are  four  towns.  A,  B,  C,  and  D,  in  a  straight 
line.  The  distance  from  B  to  C  is  one-fifth  of  the  distance 
from  A  to  B,  and  the  distance  from  C  to  D  is  equal  to 
twice  the  distance  from  A  to  C.  The  whole  distance  from 
A  to  D  is  72  miles.  Required  the  distance  from  A  to  B, 
B  to  C,  and  C  to  D. 

MODES    OF    REPRESENTING    THE 
OPERATIONS. 

ADDITION. 

2.  Illus.  1.  The  sum  of  y  -\-y  -\-y  -\-  etc.  written  seven 
times  is  7y. 

Illus.  2.  The  sum  of  m  +  m  4-  m  -f  etc.  written  x  times 
is  xm. 

The  7  and  x  are  called  the  coefficients  of  the  number 
following. 

The  coefficient  is  the  number  which  shows  how  many 
times  the  number  following  is  taken  additively.  If  no 
coefficient  is  expressed,  one  is  understood. 

Read  each  of  the  following  numbers,  name  the  coeffi- 
cient, and  state  what  it  shows  : 

6a,  2y,  3x,  ax,  5m,  9c,  xy,  mn,  lOz,  a,  25n,  x,  llxy. 


NOTATION.  81 

Illus.  3.     If  John  has  x  marbles,  and  his  brother  gives 
him  5  marbles,  how  many  has  he  ? 

Illus.  4.     If  Mary  has  x  dolls,  and  her  mother  gives  her 
y  dolls,  how  many  has  she  ".' 

Addition  is    expressed   by   coefficient   and   hy    sign 

plus  (  +  ). 

When  use  the  coefficient  ?   When  the  sign  ? 

Exercise  lO. 

1.  Charles  walked  x  miles  and  rode  9  miles.  How 
far  did  he  go  ? 

2.  A  merchant  bought  a  barrels  of  sugar  and  p  barrels 
of  molasses.     How  many  barrels  in  all  did  he  buy  ? 

3.  What  is  the  sum  of  6  +  6  +  6  -f  etc.  written  eight 
times  ? 

4.  Express  the  sum  of  x  and  y. 

5.  There  are  c  boys  at  play,  and  5  others  join  them. 
How  many  boys  are  there  in  all  ? 

6.  What  is  the  sum  oi  x -\- x -{- x  •\-  etc.  written  d  times  ? 

7.  A  lady  bought  a  silk  dress  for  m  dollars,  a  muff 
for  I  dollars,  a  shawl  for  v  dollars,  and  a  pair  of  gloves 
for  c  dollars.     What  was  the  entire  cost  ? 

8.  George  is  x  years  old,  Martin  is  y,  and  Morgan  is 
z  years.     What  is  the  sum  of  their  ages  ? 

9.  What  is  the  sum  of  m  taken  h  times  ? 

10.    If  d  is  a  whole  number,  what   is   the   next   larger 
number? 


32  A   FIRST  BOOK  IN  ALGEBRA. 

11.  A  boy  bought  a  pound  of  butter  for  y  cents,  a  pound 
of  meat  for  z  cents,  and  a  bunch  of  lettuce  for  s  cents. 
How  much  did  they  all  cost  ? 

12.  What  is  the  next  whole  number  larger  than  m  ? 

13.  What  is  the  sum  of  x  taken  y  times  ? 

14.  A  merchant  sold  x  barrels  of  flour  one  week,  40  the 
next  week,  and  a  barrels  the  following  week.  How  many 
barrels  did  he  sell  ? 

15.  Find  two  numbers  whose  sum  is  74  and  whose 
difference  is  18. 

SUBTRACTIOlSr. 

3.  Illus.  1.  A  man  sold  a  horse  for  $225  and  gained 
$  75.     What  did  the  horse  cost  ? 

Illus.  2.  A  farmer  sold  a  sheep  for  m  dollars  and  gained 
y  dollars.     What  did  the  sheep  cost  ?       Ans.  m  —  y  dollars. 

Subtraction  is  expressed  by  the  sign  tninus  (-). 

Illus.  3.  A  man  started  at  a  certain  point  and  traveled 
north  15  miles,  then  south  30  miles,  then  north  20  miles, 
then  north  5  miles,  then  south  6  miles.  How  far  is  he 
from  where  he  started  and  in  which  direction  ? 

Illus.  4.  A  man  started  at  a  certain  point  and  traveled 
east  X  miles,  then  west  b  miles,  then  east  m  miles,  then  east 
y  miles,  then  west  z  miles.  How  far  is  he  from  where  he 
started  ? 

We  find  a  difficulty  in  solving  this  last  example, 
because  we  do  not  know  just  how  large  x,  6,  m,  ?/,  and  z 
are  with    reference  to  each  other.      This  is  only  one 


NOTATION.  33 

example  of  a  large  class  of  problems  which  may  arise, 
in  which  we  find  direction  east  and  west,  north  and 
south;  space  before  and  behind,  to  the  right  and  to 
the  left,  above  and  below ;  time  past  and  future  ;  money- 
gained  and  lost;  everywhere  these  opposite  relations. 
This  relation  of  oppositeness  must  be  expressed  in  some 
way  in  our  representation  of  numbers. 

In  algebra,  therefore,  numbers  are  considered  as 
increasing  from  zero  in  opj)Osite  directions.  Those  in 
one  direction  are  called  Positive  Numbers  (or  -f  num- 
bers) ;  those  in  the  other  direction  Negative  Numbei-s 
(or  —  numbei's). 

In  lUus.  4,  if  we  call  direction  east  positive,  then 
direction  west  will  be  negative,  and  the  respective 
distances  that  the  man  ti*aveled  will  be  +  2:,  —  ^,  +  wi, 
4-  y-,  and  —  z.  Combining  these,  the  answer  to  the 
problem  becomes  x  —  h-\-m-\-y  —  z.  If  the  same  analysis 
be  applied  to  Illus.  3,  we  get  15-30  +  20-1-5-6= -h 4, 
or  4  miles  north  of  starting-point. 

The  minus  sign  before  a  single  number  makes  the 
number  negative,  and  shows  that  the  number  has  a 
subtractive  relation  to  any  other  to  which  it  may  be 
united,  and  that  it  will  diminish  that  number  by  its 
value.    It  shows  a  relation  rather  than  an  operation. 

Negative  numbers  are  the  second  of  the  two  things 
referred  to  on  page  7,  the  introduction  of  which  makes 
all  the  difference  between  arithmetic  and  algebra. 


34  A   FIRST  BOOK  IN  ALGEBRA. 

Note.  —  Negative  numbers  are  usually  spoken  of  as  less  than  zero, 
because  they  are  used  to  represent  losses.  To  illustrate :  suppose  a 
man's  money  affairs  be  such  that  his  debts  just  equal  his  assets,  we 
say  that  he  is  worth  nothing.  Suppose  now  that  the  sum  of  his  debts 
is  $  1000  greater  than  his  total  assets.  He  is  worse  off  than  by  the 
first  supposition,  and  we  say  that  he  is  worth  less  than  nothing.  We 
should  represent  his  property  by  -  1000  (dollars). 

Exercise  11. 

1.  Express  the  difference  between  a  and  b. 

2.  By  how  much  is  b  greater  than  10  ? 

3.  Express  the  sum  of  a  and  b  diminished  by  c. 

4.  Write  five  numbers  in  order  of  magnitude  so  that  a 
shall  be  the  middle  number. 

5.  A  man  has  an  income  of  a  dollars.  His  expenses 
are  b  dollars.     How  much  has  he  left  ? 

6.  How  much  less  than  c  is  8  ? 

7.  A  man  has  four  daughters  each  of  whom  is  3  years 
older  than  the  next  younger.  If  x  represent  the  age 
of  the  oldest,  what  will  represent  the  age  of  the  others  ? 

8.  A  farmer  bought  a  cow  for  b  dollars  and  sold  it  for 
c  dollars.     How  much  did  he  gain  ? 

9.  How  much  greater  than  5  is  x? 

10.  If  the  difference  between  two  numbers  is  9,  how 
may  you  represent  the  numbers  ? 

11.  A  man  sold  a  house  for  x  dollars  and  gained  f  75. 
What  did  the  house  cost  ? 


NOTATION.  85 

12.  A  man  sells  a  carriage  for  m  dollars  and  loses  x 
dollars.     What  was  the  cost  of  the  carriage  ? 

13.  I  paid  c  cents  for  a  pound  of  butter,  and  /  cents  for 
a  lemon.  How  much  more  did  the  butter  cost  than  the 
lemon  ? 

14.  Sold  a  lot  of  wood  for  h  dollars,  and  received  in 
payment  a  barrel  of  flour  worth  e  dollars.  How  many 
dollars  remain  due  ? 

15.  A  man  sold  a  cow  for  /  dollars,  a  calf  for  4  dollars, 
and  a  sheep  for  m  dollars,  and  in  payment  received  a 
wagon  worth  x  dollars.     How  much  remains  due  ? 

16.  A  box  of  raisins  was  bought  for  a  dollars,  and  a 
firkin  of  butter  for  h  dollars.  If  both  were  sold  for  c 
dollars,  how  much  was  gained  ? 

17.  At  a  certain  election  1065  ballots  were  cast  for  two 
candidates,  and  the  winning  candidate  had  a  majority  of 
207.     How  many  votes  did  each  receive  ? 

18.  A  merchant  started  the  year  with  m  dollars;  the 
first  month  he  gained  x  dollars,  the  next  month  he  lost  y 
dollars,  the  third  month  he  gained  h  dollars,  and  the  fourth 
month  lost  z  dollars.  How  much  had  he  at  the  end  of  that 
month  ? 

19.  A  man  sold  a  cow  for  $80,  and  gained  c  dollars. 
What  did  the  cow  cost  ? 

20.  If  the  sum  of  two  numbers  is  60,  how  may  the 
numbers  be  represented  ? 


36  A   FIRST  BOOK  IN  ALGEBRA. 

MULTIPLICATION. 

4.   Illus.  1.     4:'  5  -a-b-c,     7x6,    x  x  y. 
Illus.  2.     abc,     xy,     amx. 
Illus.  3.     x-x  =  xx  =  oi^. 

These  two  are  read  "x  second  power,"  or  "a;  square," 
and  'fx  third  power,"  or  "a;  cube,"  and  are  called  powers 
of  x. 

A  power  is  a  product  of  like  factors. 

The  2  and  the  3  are  called  the  exponents  of  the 
power. 

An  exponent  is  a  number  expressed  at  the  right  and 
a  little  above  another  number  to  show  how  many  times 
it  is  taken  as  a  factor. 

Multiplication  is  expressed  (1)  by  signs f  i.e.  the 
dot  and  the  cross;  (2)  by  writing  the  factors  suc- 
cessively; {3)  by  exponent. 

The  last  two  are  the  more  common  methods. 
When  use  the  exponent?     When  write  the  factors 
successively  ? 

Exercise  12. 

1 .  Express  the  double  of  x. 

2.  Express  the  product  of  a?,  y,  and  z. 

3.  How  many  cents  in  x  dollars  ? 


NOTATION.  37 

4.    Write  a  times  h  times  c. 

6.    What  will  a  quarts  of  cherries  cost  at  d  cents  a 
quart  ? 

6.  If  a  stage  coach  goes  h  miles  an  hour,  how  far  will  it 
go  in  m  hours  ? 

7.  In  a  cornfield  there  are  x  rows,  and  a  hills  in  a  row. 
How  many  hills  in  the  field  ? 

8.  Write  the  cube  of  x. 

9.  Express  in  a  different  way  ay.ay.ay,axaxaxa 
X  a  X  a. 

10.  Express  the  product  of  a  factors  each  equal  to  d. 

11.  Write  the  second  power  of  a  added  to  three  times 
the  cube  of  m. 

12.  Express  x  to  the  power  2m,  plus  x  to  the  power  m. 

13.  What  is  the  interest  on  x  dollars  for  m  years  at  6  %? 

14.  In  a  certain  school  there  are  c  girls,  and  three  times 
as  many  boys  less  8.  How  many  boys,  and  how  many  boys 
and  girls  together? 

15.  If  X  men  can  do  a  piece  of  work  in  9  days,  how 
many  days  would  it 'take  1  man  to  perform  the  same  work  ? 

16.  How  many  thirds  are  there  in  x  ? 

17.  How  many  fifths  are  there  in  6  ? 

18.  A  man  bought  a  horse  for  x  dollars,  paid  2  dollars  a 
week  for  his  keeping,  and  received  4  dollars  a  week  for  his 
work.  At  the  expiration  of  a  weeks  he  sold  him  for  m 
dollars.     How  much  did  he  gain  ? 


38  A   FIRST  BOOK  IN  ALGEBRA. 

19.  James  has  a  walnuts,  John  twice  as  many  less  8, 
and  Joseph  three  times  as  many  as  James  and  John  less  7. 
How  many  have  all  together  ? 

DIVISION^. 

5.   Illus.     a-i-b,     -• 

y 

Division  is  expressed  by  the  division  sign,  and  by 
ivriting  the  numbers  in  the  fractional  form. 

Exercise  13. 

1.  Express  five  times  a  divided  by  three  times  c. 

2.  How  many  dollars  in  y  cents  ? 

3.  How  many  books  at  a  dimes  each  can  be  bought  for 
X  dimes  ? 

4.  How  many  days  will  a  man  be  required  to  work  for 
m  dollars  if  he  receive  y  dollars  a  day  ? 

5.  X  dollars  were  given  for  b  barrels  of  flour.     What 
was  the  cost  per  barrel  ? 

6.  Express  a  plus  b,  divided  by  c. 

7.  Express  a,  plus  b  divided  by  c.  . 

8.  A  man   had  a  sons  and   half  as   many  daughters. 
How  many  children  had  he  ? 

9.  If  the  number  of  minutes  in  an  hour  be  represented 
by  X,  what  will  express  the  number  of  seconds  in  5  hours  ? 

10.  A  boy  who  earns  b  dollars  a  day  spends  x  dollars  a 
week.     How  much  has  he  at  the  end  of  3  weeks  ? 


NOTATION.  89 

11.  A  can  perform  a  piece  of  work  in  x  days,  B  in  y 
days,  and  C  in  2  days.  Express  the  part  of  the  work  that 
each  can  do  in  one  day.  Express  what  part  they  can  all  do 
in  one  day. 

12.  How  many  square  feet  in  a  garden  a  feet  on  each 
side? 

13.  A  money  drawer  contains  a  dollars,  h  dimes,  and  c 
quarters.     Express  the  whole  amount  in  cents. 

14.  a;  is  how  many  times  y  ? 

15.  If  m  apples  are  worth  n  chestnuts,  how  many  chest- 
nuts is  one  apple  worth  ? 

16.  Divide  30  apples  between  two  boys  so  that  the 
younger  may  have  two-thirds  as  many  as  the  elder. 


>>»«c 


ALGEBRAIC   EXPRESSIONS. 

6.  Illus.     a,     —  c,     b-\-S,     m  —  x-i-2c\ 

An  ulgrebraic  expression  is  any  representation  of  a 
number  by  algebraic  notation. 

7.  Illus.  1.     -3a%    2x-\-a^z^ -ocl\ 

—  3a-b  is  called  a  term,  2x  is  a  term,  -f-  aV  is  a  term, 
—  5  d*  is  a  term. 

A  term  is  an  algebraic  expression  not  connected  with 
any  other  by  the  sign  plus  or  minus,  or  one  of  the  parts 
of  an  algebraic  expression  with  its  own  sign  plus  or 


40  A   FIRST  BOOK  IN  ALGEBRA. 

minus.     If  no  sign  is  written,  the  plus  sign  is  under- 
stood.    By  what  signs  are  terms  separated  ? 

Illus.  2.  a-bc  3a^y^ 

—  7  a^bc  —  a^2/^ 

5a^bc  i^V 

The  terms  in  these  groups  are  said  to  be  similar. 

Illus.  3.  a^y.  xy  a?y 

Sa'b  Sx'y  Sab 

The  terms  of  these  groups  are  said  to  be  dissimilar. 

Similar  terms  are  terms  having  the  same  letters 
affected  by  the  same  exponents. 

Dissimilar  terms  are  terms  which  differ  in  letters 
or  exponents,  or  both. 

How  may  similar  terms  differ  ? 

Illus.  4.   abxy  ....  fourth  degree .  .  .  .  7x^y^ 
a?  .  .  .  .  third  degree    ....  abc 
Sxy  .  .  .  .  second  degree    .  .  .  a^ 
2a^bx^  ....  sixth  degree  .  .  .  .  Aa^b 

The  degree  of  a  term  is  the  number  of  its  literal 
factors.  It  can  be  found  by  taking  the  sum  of  its 
exponents. 

Illus.  5.  2a^ 

-a^y 

How  do  these  terms  compare  with  reference  to  degree  ? 
They  are  called  homogeneous  terms. 

What  are  homogeneous  terms? 


NOTATION.  41 

8.   Illus.   3ary  called  a  monomial. 

7a?  —  2xy  ) 

«   .       ,         ^    „  !■    called  polynomials. 

A  monomial  is  an  algebraic  expression  of  one  term. 

A  polynomial  is  an  algebraic  expression  of  more 
than  one  term. 

A  polynomial  of  two  terms  is  called  a  binomial,  and 
one  of  three  terms  is  called  a  trinomial. 

The  degree  of  an  algebraic  expression  is  the  same 
as  the  degree  of  its  highest  term.  What  is  the  degree 
of  each  of  the  polynomials  above?  What  is  a  homo- 
geneous polynomial  ? 

Exercise  14. 

1.  Write  a  polynomial  of  five  terms.  Of  what  degree 
is  it? 

2.  Write  a  binomial  of  the  fourth  degree. 

3.  Write  a  polynomial  with  the  terms  of  different 
degrees. 

4.  Write  a  homogeneous  trinomial  of  the  third  degree. 

5.  Write  two  similar  monomials  of  the  fifth  degree 
which  shall  differ  as  much  as  possible. 

6.  Write  a  homogeneous  trinomial  with  one  of  its 
terms  of  the  second  degree. 

7.  Arrange  according  to  the  descending  powers  of  a : 
-  80a'»6«  -f  60a*6»  +  108a6*  -f  48a*6  +  3a«  -  27 6«  -  OOa^ft*. 

What  name  ?    What  degree  ? 


42  A   FIRST  BOOK  IN  ALGEBRA. 

8.  Write  a  polynomial  of  the  fifth  degree  containing 
six  terms. 

9.  Arrange  according  to  the  ascending  powers  of  x: 

15*22/3  -\-  7 x^  -  3xf  -  eOx'y  +  y^  -\-  21a^y\ 

What  name  ?     What  degree  ?     What  is   the  degree   of 
each  term  ? 

When  a  =  l,   5  =  2,  c  =  3,   d  =  4,  x  =  0,   ?/ =  8,  find 
the  value  of  the  following : 

10.  2a +  36  4- c. 

11.  5b-{-3a-2c  +  6x. 

12.  6bc  —  3ax  +  2xb  —  5ac-}-2cx. 

13.  3bcd-{- 5cxa  —  7xab -\- abc. 

14.  2c^-^Sb^  +  4:a\ 

15.  ^ a^c  —  6^  _  c^  _  ^abc^. 

16.  2a -5       ^^^ 


a  +  6 

17.    2bc  —  ^c^-\-Sab  —  2a  —  x-\--^jbx. 
a^bx  +  aft^c  -f  a6c^  +  .Tac^ 


18. 


abc 


19.  Henry  bought  some  apples  at  3  cents  apiece,  and 
twice  as  many  pears  at  4  cents  apiece,  paying  for  the 
whole  66  cents.     How  many  of  each  did  he  buy  ? 

20.  Sarah's  father  told  her  that  the  difference  between 
two-thirds  and  five-sixths  of  his  age  was  6  years.  How 
old  was  he  ? 


OPERATIONS. 


ADDITION. 

9.  In  combining  numbers  in  algebra  it  must  always 
be  borne  in  mind  that  negative  number  are  the  oppo- 
site of  positive  numbers  in  their  tendency. 

Illus.  1.  Sax  —    Ih-y 

5ax  -    Sb'y 

2ax  —   Ab-y 

10  ax  -Ub^y 

To  add  similar  terms  with  Wee  signs,  add  the 
coefficients,  annex  tJie  common  letters,  and  prefix 
tJie  common  sign. 

Illus.  2. 


Sa^ft 

3x2/ 

-Sa^b 

8x2/ 

-4.a% 

-5x2/ 

6a*6 

-IxY 

4a*6  -    xy 

To  add  similar  terms  with  unlike  signs,  add  the 

coefficients  of  the  plus  terms,  add  the  coefficients  of 

tlie   minus   terms,   to   the  difference   of  these  sums 

annex  the  coTnmon  letters,  and  prefix  the  sign  of 

the  greater  sum. 

43 


44  A    FIRST  BOOK  IN  ALGEBRA. 

Illus.  3. 


a 

2x 

b 

-5y 

c 

-Sa 

a-{-b-\-c  2x  —  5y  —  Sa 

To  add  dissimilar  terms,  write  the  terms  succes- 
sively, each  with  its  own  sign. 

Illus.  4.  2ab—    3ax'-\-2a'x 

—  Sab  —      ax^  —  5  a-x  +  ax^ 
12ab-^10ax'-6a'x 

6ab-\-    6ax^ —  9a'x-\- ax^ 

To  add  polynomials,  add  the  terms  of  which  the 
polynomials  consist,  and  unite  the  results. 

Exercise  15. 

Find  the  sum  of  : 

1.  3x,  5x,  x,  ix,  11a;. 

2.  Bab,  6ab,  ab,  IS ab. 

3.  —Sax^,  —5a3P^,   —9ax^,  —ax^. 

4.  —x,—5x,   —11a;,   —25  a;. 

5.  -2a',  5a%  Sa%   -7a',  lla\ 

6.  2abc',  -5abc%  ahc\   -Sabc\ 

7.  5x',  Sab,   -2ab,   -4:X%  Bab,   -2x\ 

8.  5ax,   —36c,  —2ax,  lax,  be,  —26c. 
Simplify  : 

9.  4,a'-5a'-Sa'-7a\ 

10.  a;^  +  5a^6-7a6-2ar^  +  10a6  +  3a^6. 

11.  ^a  —  la-{-^a-\-a. 


OPERATIONS.  46 

12.  lh-\h-2h-\h  +  ih->rh. 

13.  A  lady  bought  a  ribbon  for  m  cents,  some  tape  for 
d  cents,  and  some  thread  for  c  cents.  She  paid  x  cents  on 
the  bill.     How  much  remains  due  ? 

14.  A  man  travels  a  miles  north,  then  x  miles  south, 
then  5  miles  further  south,  and  then  y  miles  north.  How 
far  is  he  from  his  starting  point  ? 

Add: 

16.  o  4- 26  4- 3c,  5a -1-36 -he,  c  —  a  —  h. 

16.  x-\'y  —  z,  x  —  y  —  z,  y  —  x-\-z. 

17.  x-\-2y-'Sz-\-a,  2x-3y -{-z- 4a,  2a-3x-\-y-Z' 

18.  x'+Sx'-x+S,    4x2-5ar^-H3-4a;,    3x+6a^-33^-\-9. 

19.  ca—bc  +  c^y  ab  +  h^  —  cay  a^  —  ab-\- be. 

20.  3a-»-a"»-*-l,  3 a— ' -f  1  - 2 a"»,  a"*- 2a"-' -f  1. 

21.  5  a^  -  16  a*6- 11  a-62c+ 13  a6,     -2a*-|-4a*6-|-12a262c 

-  10  a6,     6  a*  -  a*6  -  6  a^b-c  +  10  ab,     -  10  a^  +  8  a^6  -|-  a'^b^c 

-  6a6,     a*  +  5a^6  -{-6a'b'c  -  7a6. 

22.  15af»-|-35a^4-3x-h7,  7 x" -{■lox-Ux' +  9,  9x-10 

23.  9aj»i/-6a;y  4-«y-25a;t/^,      -  22 ar'y3_  3 2y_  9 ^j^ 

-  3a;y,     5ary  +  x^y  +  21a^/+  203^. 

24.  x  —  y  —  z  —  a  —  b,  x-\-y-\-z  +  a-{-b,  x-^y+z  +  a—b, 
x-\-y  —  z  —  a  —  b,  x-\-y-{-z  —  a  —  b. 

26.    a-c  -\.b^c-\-<f-  abc  -b(^-ac*,        a^b  +  6*  -f-  6c»  -  ab^ 

-  bh  —  ahCy     a*  -|-  ab^  +  0^  —  a^b  —  abc  —  a^c. 


46  A   FIRST  BOOK  IN  ALGEBRA. 

26.  A  regiment  is  drawn  up  in  m  ranks  of  b  men  each, 
and  there  are  c  men  over.     How  many  men  in  the  regiment  ? 

27.  A  man  had  x  cows  and  z  horses.  After  exchanging 
10  cows  with  another  man  for  19  horses,  what  will  repre- 
sent the  number  that  he  has  of  each  ? 

28.  In  a  class  of  52  pupils  there  are  8  more  boys  than 
girls.     How  many  are  there  of  each  ? 

What  is  the  sum  of  two  numbers  equal  numerically 
but  of  opposite  sign  ?  How  does  the  sum  of  a  positive 
and  negative  number  compare  in  value  with  the  positive 
number?  with  the  negative  number?  How  does  the 
sum  of  two  negative  numbers  compare  with  the  num- 
bers ?  Illustrate  the  above  questions  by  a  man  traveling 
north  and  south. 

SUBTRACTION. 

10.   How  is  subtraction  related  to  addition?     How 
are  opposite  relations  expressed? 
Given  the  typical  series  of  numbers 
-  4  a,    -  3  a,    —2  a,    -  a,    -  0,    a,    2  a,    3  a,    4:  a,    6  a. 

What  must  be  added  to  2  a  to  obtain  5a?  What  then 
must  be  subtracted  from  5a  to  obtain  2a?     5a  — 3a  =  ? 

What  must  be  added  to  —  3  a  to  obtain  4a?  What 
then  must  be  subtracted  from  4a  to  obtain  —3a? 
4a-7a  =  ? 


OPERATIONS.  47 

What  must  be  added  to  3a  to  obtain  —  2a?  What 
then  must  be  subtracted  from  —2a  to  obtain  3a? 
(_2a)-(-5a)=? 

What  must  be  added  to  —a  to  obtain  —4a?  What 
tlien  must  be  subtracted  from  —4a  to  obtain  —a? 
(-4a)-(-3a)=? 

Examine  now  these  results  expressed  in  another  form. 


1.    From 

5a 

To 

5a 

take 

3a 

add 

-3a 

2a 

2a 

2.   From 

4a 

To 

4a 

take 

la 

add 

-la 

-3a 

-3a 

3.    From 

-2a 

To 

-2a 

take 

-5a 

add 

5a 

3a 

3a 

4.   From 

-4a 

To 

-4a 

take 

-3a 

add 

3a 

—    a  —a 

The  principle  is  clear ;  namely, 

The  subtraction  of  any  number  gives  the  same  result 
as  the  addition  of  that  number  with  the  opposite  sign, 

Illus.  6a +  36—    c 

—  4a+    6  — 5c 
l6a  +  26  +  4c 
To  subtract  one  number  from  another,  consider  the 
sign  of  the  subtrahend  changed  and  add. 


48  A   FIRST  BOOK  IN  ALGEBRA. 

What  is  the  relation  of  the  minuend  to  the  subtra- 
hend and  remainder?  What  is  the  relation  of  the 
subtrahend  to  the  minuend  and  remainder? 

Exercise  16. 

1.  From  5  a^  take  3  a^. 

2.  Froni  7a-b  take  —  5a^6. 

3.  Subtract  7xy^  from  —2xy^. 

4.  From  —  3a;"*2/  take  —Ix'^y. 

5.  Subtract  ^ax  from  8ic^. 

6.  From  5  xy  take  —  7  hy. 

7.  What  is  the  difference  between  4  a""  and  2  a'"  ? 

8.  From  the  difference  between  ba^x  and  —Sa^x  take 
the  sum  of  2a^x  and  —Sa^x. 

9.  From  2a  + 6  + 7c  take  5a +  26  — 7c. 

10.  From  9x  — 4:y -\-3z  tsike  5x  —  3y -{-z. 

11.  Subtract  3x'-x'+7x-U  from  lla;*-2ar'+3a;2_8a;. 

12.  From  10 a'b' -{- 15 ab' -{- S a^b  take  -lOa'b'' +  Wab^ 
-Sa'b. 

13.  Subtract  1- a; 4-a^-3a:»  from  a^'  —  l  +  ar' -a;. 

14.  From  aj'"-2a^'"  +  a^"*  take  2a^'"-a^"»-a;"'. 

15.  Subtract  a^'*  +  a"a;«  +  a;2«  from  Sa^"  -  17a"a;"  -Sa^\ 

16.  From|a2-fa-l  take -|a2  +  a-i. 

17.  Fromar^  +  3aJ2^takes?'4-2a;V  +  3a;3y2_2a;y^  +  y. 


OPERATIONS.  49 

18.  From  x  take  y  —  a. 

19.  From  6a'4-4a-f  7  take  the  sum  of  2a^  +  4a2-f-9 
and  4a*  —  a^  -f  4a  —  2. 

20.  Subtract  3a;--7af*4-5a:^  from  the  sum  of  2  +  8ar  —  ar* 
and2ar'-3ar  +  x-2. 

21.  What  must  be  subtracted  from  15f/^+2r'-f  41/2^— Sz'a; 
—  2  an/*  to  leave  a  remainder  of  6af'— 127/^+42^— 2  a;/ +6  z^x? 

22.  How  much  must  be  added  to  ar*  — 4.x2  + 16a;  to  pro- 
duce ar"*  -f  64  ? 

23.  To  what  must  4a-— 66- -f  86c  — 6a6  be   added  to 
produce  zero  ? 

24.  From  what  must  2a;*  — 3a;^-|-2a;  — 5  be  subtracted 
to  produce  unity  ? 

^  25.   What  must  be  subtracted  from  the  sum  of  3a'-f  7  a 
+  1  and  2a^— 5a— 3  to  leave  a  remainder  of  2a*— 2a''— 4? 

26.  From  the  difference  between  10a*64-8a6-— Sa^^'^— ft** 
and  oa-6-6a6*-7a262  take  the  sum  of  10a'6'+15a62+8a-6 

27.  What  must  be  added  to  a  to  make  h  ? 

28.  By  how  much  does  3a;  — 2  exceed  2 a;4-l? 

29.  In  y  years  a  man  will  be  40  years  old.     What  is  his 
present  age  ? 

30.  How  many  hours  will  it  take  to  go  23  miles  at  a 
miles  an  hour  ? 


50  A   FIRST  BOOK  IN  ALGEBRA. 

PARENTHESES. 

11.   Illus.  1.     5{a  +  b). 

Illus.  2.     (m -\- n)  (x -{- y) . 
Illus.  3.     x  —  (a-{-y  —  c). 

The  parenthesis  indicates  that  the  numbers  enclosed  are 
considered  as  one  number. 

Eead  each  of  the  above  illustrations,  state  the  operations 
expressed,  and  show  what  the  parenthesis  indicates. 

Write  the  expressions  for  the  following : 

1.  The  sum  of  a  and  b,  multiplied  by  a  minus  b. 

2.  c  plus  d,  times  the  sum  of  a  and  b,  —  the  whole 
multiplied  by  x  minus  y. 

3.  The  sum  of  a  and  b,  minus  the  difference  between 
two  a  and  three  b. 

4.  {x  —  y)  -\-  {x  —  y)  -{-  (x—y)  -\-  etc.,  written  a  times. 

5.  The  sum  of  a  -f  6  taken  seven  times. 

6.  There  are  in  a  library  m  -f  n  books,  each  book  has 
c  —  d  pages,  and  each  page  contains  x-\-y  words.  How 
many  words  in  all  the  books  ? 

Illus.  4.     a-{-(b  —  c  —  x)  =  a-{-b  —  c  —  x. 
(By  performing  the  addition.) 

Illus.  5.     a  +  c  —  d  +  e  =  a  +  (c  —  c?  +  e). 

Any  number  of  terms  may  be  removed  from  a 
parenthesis  preceded  by  the  plus  sign  without  change 
in  the  terms. 


OPERATIONS.  61 

And  conversely, 

Any  number  of  terms  fnay  be  enclosed  in  a  paren- 
thesis preceded  by  the  plus  sign  without  change  in 
tlie  terms. 

Illus.  6.     x  —  {y-\-z  —  c)  =  x  —  y  —  Z'\-c. 
(By  performing  the  subtraction.) 

Illus.  7.     a  —  h  —  c-\-d  =  a  —  {h-\-c—d). 

Any  number  of  terms  may  be  removed  from  a 
parenthesis  2>**^ceded  by  the  minus  sign  by  changing 
the  sign  of  each  term. 

And  conversely, 

^  Any  number  of  terms  may  be  enclosed  in  a  paren- 
thesis preceded  by  the  minus  sign  by  changing  the 
sign  of  each  term, 

Exercise  17. 

Remove  the  parentheses  in  the  following : 

1.  x  +  (a  +  h)^-y-{-{c-d)  +  {x-y). 

2.  a-f  (6-c)-6  4-(a  +  c)4-(c-a). 

3.  a^h  -  (a^  +  6»)  -a?-  (ab^  -  a^b)  -  {b^  -  a'). 

4.  xy-{3^  +  f)-f-(x^-2xy)-(f-x'y 

5.  (a  -I-  6  —  c)  —  (a  —  6  +  c)  +  (6— a— c)  —  {c  —  a  —  b). 

6.  (a;  -  2/  +  2)  +  (a;  +  y  +  2)  -  (y  4-  35  +  z)  -  (z  +  a;  +  y) . 

7.  a-{Sb-2c  +  a)-(2b-a-c)-{b-c  +  a), 

8.  ia-ic-(J6-ic)-(a+ic-i6)-(J6-Jc-ia). 


62  A   FIRST  BOOK  IN  ALGEBRA. 

In  each  of  the  following  enclose  the  last  two  terms  in 
a  parenthesis  preceded  by  a  plus  sign : 

9.  x  —  y-\-2c  —  d. 

10.  2a~  +  3a^x-ab'  +  by\ 

11.  10  m^  -\-31m^  -20m  -21, 

12.  ax^  —  a:^-\-2x  —  2 ax^.  ■ 

In  each  of  the  following  enclose  the  last  three  terms 
in  a  parenthesis  preceded  by  a  minus  sign : 

13 .  a'^  -\-  o?x  +  a^x"^  —  ax^  —  4  «''. 

14.  a^  +  a^-Ga^-f  a  +  3. 

15.  6a-^-17a2x  +  14ax2-3x^. 

16.  ax^ -\- 2  av? -\- ax -\- 2 a. 

17.  A  man  pumps  x  gallons  of  water  into  a  tank  each 
day,  and  draws  off  y  gallons  each  day.  How  much  water 
will  remain  in  the  tank  at  the  end  of  five  days  ? 

18.  Two  men  are  150  miles  apart,  and  approach  each 
other,  one  at  the  rate  of  x  miles  an  hour,  the  other  at  the 
rate  of  y  miles  an  hour.  How  far  apart  will  they  be  at  the 
end  of  seven  hours  ? 

19.  Eight  years  ago  A  was  x  years  old.  How  old  is  he 
now? 

20.  A  had  x  dollars,  but  after  giving  $35  to  B  he  has 
one-third  as  much  as  B.     How  much  has  B  ? 


OPERATIONS.  63 


MULTIPLICATION. 


12.   Illus.  1.     8=r2.2.2  a^^a-a^a 

6  =  2-3  6*  =  6. 6 


48  =  2. 2. 2. 2. 3  a^h^^^a^  a-a-h-h 


Illus.  2.  2d^b'^c 


6a«6V 


In  arithmetic  you  learned  that  multiplication  is  the  addition  of 
equal  numbers,  that  the  multiplicand  expresses  one  of  those  equal 
numbers,  and  the  multiplier  the  number  of  them.  In  algebra  we 
have  negative  as  well  as  positive  numbers.  Let  us  see  the  effect  of 
this  in  multiplication.     We  have  four  possible  cases. 

1.  Multiplication  of  a  plus  number  by  a  plus  number. 

+  7 
Illds.  +  4    Tliis  must  mean  four  sevens,  or  28. 

2.  Multiplication  of  a  minus  number  by  a  plus  number. 

-7 
Illus.  -f  4    This  must  mean  four  minus-sevens,  or  —  28. 

3.  Multiplication  of  a  plus  number  by  a  minus  number. 

+  7 
Illus.  -4    This  must  mean  the  opposite  of  what  +  4  meant  as  a 

multiplier.    Plus  four  meant  add,  minus  four  must  mean  subtract. 

Subtracting  four  sevens  gives  —  28. 

4.  Multiplication  of  a  minus  number  by  a  minus  number. 

-7 
Illus.   —  4    This  must  mean  sM6<rac( /our  wnius-screns,  or  28. 

Illus.  3.         +6  —b  +6  —b 

-\-a  4-a  —a  — a 

-fad  —ab  —ab  +cU> 


54  A    FIRST  BOOK  IN  ALGEBRA. 

To  multiply  a  jnonoinial  by  a  monomial,  multi- 
ply the  coefficients  together  for  the  coefficient  of  the 
product,  add  the  exponents  of  like  letters  for  the 
exponent  of  the  same  letter  in  the  product,  and 
give  the  product  of  two  numbers  having  like  signs 
the  plus  sign,  having  unlike  signs  the  minus  sign. 

Exercise  18. 

Find  the  product  of : 

1.  5a;  and  7c.  5.  —  3 a^^y, —2 aa;,  and  3 c^/^. 

2 .  51 C2/  and  —  xa.  6 .  5  a^,  —  3  6c^,  and  —  2  dbc. 

3.  3  a;^?/ and  7  aa;/.  7.  l^y:^y'^, —^xz,  diXidi  ^^yz-. 

4.  ba'hc  and  2aW(?.  8.  20a^h^,  lah\  and  -\hc\ 

10.  _- fa^d^,  ic^, -fac,  and-fd^c. 

11.  In  how  many  days  will  a  boys  eat  100  apples  if 
each  boy  eats  b  apples  a  day  ? 

12.  How  many  units  in  x  hundreds  ? 

13.  If  there  are  a  hundreds,  b  tens,  and  c  units  in  a 
number,  what  will  represent  the  whole  number  of  units  ? 

14.  If  the  difference  between  two  numbers  is  7,  and  one 
of  the  numbers  is  x,  what  is  the  other  number  ? 

13.   Illus.  1.  a  —  b-^c 

X 


ax  —  bx-\-  ex 


OPERA  TIONS.  66 

To  multiply  a  polynomidl'  by  a  monoTnial,  multi- 
ply each  term  of  tJie  polynom,ial  by  the  monoTuial, 
and  add  the  results. 

Illus.  2.  x'-\-2x^-{-Sx 

33r-2x  +1 

■-2x*-Ax^-6a^ 

Q^  +  2a^'\-3x 


3x^  +  ^x*  +  Gx^-  4a*  +  3a; 


To  m,ultiply  a  polynomial  by  a  polynomial,  multi- 
ply the  multiplicand  by  each  term  of  the  multiplier, 
and  a^d  the  products. 

How  is  the  first  term  of  the  product  obtained  ?  How  is 
the  last  term  obtained  ?  The  polynomials  being  arranged 
similarly  with  reference  to  the  exponents  of  some  number, 
how  is  the  product  arranged  ? 

Exercise  19. 

Multiply : 

1.  x^  +  xy  +  y'^hy  a^t^. 

2.  a^-ah-\-h'hy  a'b. 

3.  a? -3a-h +  h^  hy  -2ab. 

4.  8aj»  +  36a^y  +  273^by  3a^. 

5.  |a^  -\a^b-  ^a^ft*  by  faft^ 

6.  3?  "  xy  +  f  hy  X -^  y. 


56  A   FIRST  BOOK  IN  ALGEBRA. 

7.  a;<-3a,'3  +  2a;2-ic  +  l  by  a;-l. 

8.  a^-2x--]-xhy  x^-^-Sx  +  l. 

9.  xy -^  mn  —  xm  —  yn  hy  xy  —  mn -\- xm  —  yn. 

10.  x'-a^  +  x--x-{-l\)y2  +  dx-\-2s^  +  a^. 

11.  ce-a'h  +  a'W-a'h^^ah^-h'hy  a^h. 

12.  x'^  —  xy  -{-  y-  —  yz  -\-  z-  —  xz  hy  X  -\-  y  -\-  z. 

13.  x^  -{-  x^y  -[-  x^'y^  +  x^y^  +  a^2/^  +  ic^/^  -^-y'^hy  x  —  y. 

14.  ic*  —  4aV  +  4a^  by  cu^  +  4aV  +  4a^ 

15.  a»-3ay4-2/^by  a^  +  3ay  +  2/^. 

16.  a;^  +  10.-c  +  12  +  9cc2  +  3x3by-2aj  +  a;2-l.     , 

17.  3a;2-2  +  a.'^-3x  +  6ic4by-2  +  a;2-3a;. 

18.  If  05  represent  the  number  of  miles  a  man  can  row 
in  an  hour  in  still  water,  how  far  can  the  man  row  in 
5  hours  down  a  stream  which  flows  y  miles  an  hour  ?  How 
far  up  the  same  stream  in  4  hours  ? 

19.  A  can  reap  a  field  in  7  hours,  and  B  can  reap  the 
same  field  in  5  hours.  How  much  of  the  field  can  they  do 
in  one  hour,  working  together  ? 

20.  A  tank  can  be  filled  by  two  pipes  in  a  hours  and  h 
hours  respectively.  What  part  of  the  tank  will  be  filled 
by  both  pipes  running  together  for  one  hour  ? 

What  does  x  —  y  express  ?  What  two  operations  will 
give  that  result?  What  operations  will  give  4a;  as  a 
result  ? 


14.  Illus.  1. 

OPERA  Tj 

X  4-5 
X  4-3 

a;*4-5a; 

3a;  4- 15 

x*  +  8a;4-15 

X  -5 
X  -3 

fONS. 
Illus. 

Illus. 

3. 
4. 

X  4-5 
x  -3 

a;'4-5a; 
-3a; -15 

Illus.  2. 

a;2  4-2a;-15 

X  -5 
X  4-3 

x'-Bx 
-3a;  4- 15 

ar'-8a;4-15 

ar^-5a; 

3a; -15 

ar2_2a;-15 

67 


How  many  terms  in  the  product?  What  is  the  first 
term?  How  is  the  last  term  formed?  How  is  the 
coefficient  of  x  in  the  middle  term  formed  ? 

The  answers  to  the  examples  in  the  following  exercise 
are  to  be  written  directly,  and  not  to  be  obtained  by  the 
full  form  of  multiplication  : 


Exercise  20. 


Expand : 

1.  (a;  +  2)(a;4-7) 

2.  (x4-l)(a;  +  6) 

3.  (x-3)(x-4) 

4.  (x-5)(a;-2) 

5.  (x'4-5)(a;-2) 

6.  (a;4-7)(x  — 3) 


7.  (x-7)(a;  +  6). 

8.  (a; -6)  (a;  4- 5). 

9.  (x-n)(a;-2). 

10.  (a;-13)(a;-l). 

11.  (2/ 4- 7)  (2/ -9). 

12.  (a;  4- 3)  (a;  4- 17). 


68  A   FIRST  BOOK  IN  ALGEBRA. 

13.  (2/ +  2)  (2/ -15).  22.  (a-|-)(a  +  f). 

14.  (2/ +  2)  (2/ +  16).  23.  {x-l){x-i). 

15.  {a'  +  7)(a'-o).  24.  (2/ +  |)  (2/ +  i). 

16.  (a -9)  (a +  9).  25.  (3-x)(7-x). 

17.  (m2-2)(m--16).  26.  (o-x){S-x). 

18.  (63 -I- 12)  (63  _  10).  27.  (6-a;)(7  +  a;). 

19.  (x-i){x-\).  28.  (11 -a;)  (3  + a;). 

20.  (2/  +  i)(2/+i-).  29.  (0^-3)  (a; +  3). 

21.  (m  +  |)(m-i).  30.  (2/ +  5)  (2/ -  5). 

31.  Find  a  number  which,  being  multiplied  by  6,  and 
having  15  added  to  the  product,  will  equal  141. 

32.  Mr.  Allen  has  3  more  cows  than  his  neighbor.  Three 
times  his  number  of  cows  will  equal  four  times  his  neigh- 
bor's.    How  many  has  Mr.  Allen  ? 

INVOLUTION. 

15.  What  is  the  second  power  of  5  ?  What  is  the  third 
power  of  4  ? 

Involution  is  the  process  of  finding  a  power  of  a 
number. 

Illus.  1.  {5a'by=25a'b\ 

Illus.  2.  {3  xifzy  =  27  x^y'^. 

Illus.  3.  Find  by  multiplication  the  2d,  3d,  4th,  and 
6th  powers  of  +a  and  —a.  Observe  the  signs  of  the  odd 
and  of  the  even  powers. 


OPERATIONS.  69 

To  find  any  power  of  a  jyvonomial,  raise  the  coeffi- 
cient to  tJie  required  power,  multiply  tJie  exponent 
of  ea^h  letter  by  tlie  exponent  of  tlie  power,  and  give 
every  even  power  the  plus  sign,  every  odd  power  the 
sign  of  the  original  number. 

£xercise  21. 
Expand : 

1.  (a«6)*.  7.  {xyT^y.  13.  (-ISc^rfx^)^ 

2.  {xyy.  8.  {-mhidy.  14.  (-Oa^z^^s^ 

3.  {-a^hy.  9.  (-5ar'yz)3.  15.  (a^b-c^d^y. 

4.  {-a?y^y.  10.  (llc*d'V)l  16.  {-3^y:^m''ny. 

5.  (3a-?/)'.  11.  {^x^arti^y.  17.  (-|a-6c^)2. 

6.  (-7a6V)l  12.  (-Ja&*'c)l  18.  (|mnV)2. 

19.  In  how  many  days  can  one  man  do  as  much  as  b  men 
in  8  days  ? 

20.  How  many  mills  in  a  cents  ?     How  many  dollars  ? 

16.   Find  by  multiplication  the  following : 
(a +  6)-,   {a -by,  (a  +  6)»,   {a -by,  {a  +  by,  (a -by. 
Memorize  the  results. 

It  is  intended  that  the  answers  in  the  following  exercise 
shall  be  written  directly  without  going  through  the  multi- 
plication. 

Illus.  1.     {x  —  yy  =  a^  —  4x^y-^6a^}/^  —  4:Xtf  +  y*. 
Illus.  2.     {x-iy  =  a^-3a^  +  3x-l. 


60  A   FIRST  BOOK  IN  ALGEBRA. 

Illus.  3.     {2xy-{-3yy 

=  {2xyy  +  4.(2xyy(3f)  +  6{2xyy{3fy 

-{-A{2xy){Sfy  +  {3fy 
=  16  xY  +  96  x^y'  +  216  a^/  +  216  xy^  +  81 3/«. 

Exercise  22. 

Expand : 

1.  (Z  +  Xy.  9.      (c2-(^2)4  17^      (6^-1)^ 

2.  (a +  2/)'.  10.  (/ +  «")'.  18.  if  +  iy. 

3.  (aj-rt)\  11.  (a.-^^/^^)-'.  19.  (ab-2y. 

4.  (tt-m)l  12.  (a'b-cy.  20.  (.t22/-3)1 

5.  (m  +  a)'.  13.  (a'-b^cy.  21.  (1-aj)^ 

6.  (x-yy.  14.  (x22/-w^^)^-  22.  (l-.v')^ 

7.  {x^'-^-yy.  15.  (x  +  l)3.  23.  (2a;  +  3/)2. 

8.  (m^ -?/')'.  16.  (7?i-l)2.  24.  (3ab-a^yy. 

25.  (4mn3-3a^6)*.  27.    (l-^x^y. 

26.  (ia;-2/)l  28.    (a-2-3)*. 

29.  John  has  4  a  horses,  James  has  a  times  as  many  as 
John,  and  Charles  has  d  less  than  five  times  as  many 
as  James.     How  many  has  Charles  ? 

30.  A  man  bought  a  pounds  of  meat  at  a  cents  a  pound, 
and  handed  the  butcher  an  a;-dollar  bill.  How  many  cents 
in  change  should  he  receive  ? 

31.  A  grocer,  having  25  bags  of  meal  worth  a  cents  a 
bag,  sold  x  bags.     What  is  the  value  of  the  meal  left  ? 


OPERATIONS.  61 

32.  If  a  =  5,  X  =  4,  y  =  3,  find  the  numerical  value  of 

7o  11a;     _      10  y 

\\x  —  ^y     Sx  —  7y     7a  — 5x 

33.  Find  the  value  of 

a^ft  _  c^d-.  (ab  -i-cd)  (oc  -  bd)  -  bc(a^c  -  bd^) 
when  a  =  2,  6  =  3,  c  =  4,  and  d  =  0. 

Exercise  23.     (Review.) 

1.  Take  the  sum  of  x^-\-Sx-2,  2oiP-j-x^ -x  + 5,  and 
4jc»4-2ic2-7a;  +  4  from  the  sum  of  2x^  +  9a;  and  Ba^  +  Sx'. 

2.  Multiply  6*  -  2 6^  +  1  by  6*  4-  2  6»  +  1. 

3.  Simplify         lla;' +  41/2  -  (2a;?/ -  3/)  +  (2a;2- 3x2/) 
-(Sa^-5xy). 

4.  Divide  $300  among  A,  B,  and  C,  so  that  A  shall 
have  twice  as  much  as  B,  and  jB  $  20  more  than  C. 

6.    Find  two  numbers   differing   by  8  such   that   four 
times  the  less  may  exceed  twice  the  greater  by  10. 

6.  What  must  be  added  to  3  a'  — 4a^  — 4  to  produce 
5a»  +  6? 

7.  Add     ia'-ab-ib^     ^a^  +  iab-^b^,     and     -a» 
-iab-\-2b\ 

8.  Simplify  8a6V  x  (-Sa^bc')  x  {-2a^bPc). 

9.  Expand  (-Jan/V)*. 

10.  Simplify  (x  -  2)  (x  +  7)  4-  (a;  ~  8)  (x  -  5). 

11.  Expand  (2a*6-3x2/)«. 


62  A   FIRST  BOOK  IN  ALGEBRA. 

12.  What  must  be  subtracted  from  x^  —  3x^ -{-2y  —  5  to 
produce  unity  ? 

13.  Multiply  a^  +  3x-y+3xy--\-y^  by  3xy'-  -  y^-3x^y-\-x\ 

14.  Expand  {x  +  1)  (a;  -  1)  (x-  -f  1). 

15.  Add4xy-42/^    4:X^y -12  xy -{- 12  xy^ —  4.  y*, 

6xY-12xy'+6y'  and  x'-4.x'y-\-6xY-'ixy^+y\ 

16.  A  man  weighs  36  pounds  more  than  his  wife,  and 
the  sum  of  their  weights  is  317  pounds.  What  is  the 
weight  of  each  ? 

17.  A  watch  and  chain  cost  ^350.  What  was  the  cost 
of  each,  if  the  chain  cost  f  as  much  as  the  watch  ? 

18.  Simplify  3a^-2a;+l-  (x''+2x-\-3) -{2x'-6x-6). 

19.  Simplify  (a +  2 2/) 2 -(a -2 2/) I 

20.  What  is  the  value  of  1  +  ia  +  ^6  times  l  —  ^a  +  ^b? 


o>Kc 


DIVISION. 

17.   What  is  the  relation  of  division  to  multiplication? 

Illus.  30^  x2xy=z?    then   6 afy  -i-2xy  =  ? 

Division  is  the  process  by  which,  when  a  product  is 
given  and  one  factor  known,  the  other  factor  is  found. 

What  is  the  relation  of  the  dividend  to  the  divisor 
and  quotient  ?  What  factors  must  the  dividend  contain  ? 
What  factors  must  the  quotient  contain  ? 


OPERATIONS.  63 

Illus.  1.   6  a' 6V  -i-  2a»6»c«  =  3  aW(?. 

Illus.  2.    +a6  \±a^,  -ah  [+06^,  +a&  |-ad',  -  ah  \-ab\ 

From  the  relation  of  the  dividend,  divisor,  and  quotient, 
and  the  law  for  signs  in  multiplication,  obtain  the  quo- 
tients in  Illus.  2. 

To  divide  a  monomial  by  a  myonomiaZ,  divide 
the  coefficient  of  tlie  dividend  by  the  coefficient 
of  the  divisor  for  the  coefficient  of  the  quotient, 
subtract  tJie  exponent  of  each  letter  in  the  divisor 
from  the  exponent  of  the  same  letter  in  the  dividend 
for  the  exponent  of  that  letter  in  the  quotient:  if 
dividend  and  divisor  have  like  signs,  give  the  quo- 
tient the  plus  sign ;  if  unlike,  tlie  jninus  sign. 

Exercise  24. 

Divide : 

1.  15ar^by3a;.'  9.  ^ a^6*  by  J a^ft*. 

2.  39a6-by36.  '  10.  ^a^y*hj  -  ^xf. 

3.  27a«6«c  by  9  afe^c.  11.  -45a^/z  by  Qx^^z. 

4.  35 sl^y'zhy  7 x'yz.  12.  60 a^6c"  by  -  12 a^ftc^ 
6.  —  Sleazy  by  3cya^.  13.  —  f  xy  by  —  ^x*y. 

6.  121  xYz  by  -  11  yh.  14.    f  a'?/iV  by  -  \a^mn\ 

7.  -28xyz*  by  -  7xf.  15.    5mVx»  by  ^mn^x. 

8.  -  36a«dV  by  -  4a6V.         16.    4aryz«  by  - 

17.  10(x  +  y)Vby5(x  +  2/)'2. 

18.  15  (a  -  6)  V  by  3  (a  -  6)aj. 


64  A   FIRST  BOOK  IN  ALGEBRA. 

19.  Simplify  (  -  a^^/^V)  x  (  -  x^z^)  -^-  2 xYz\ 

20.  Simplify  a'b^c  x  (-  a%V) -^3(aPb(^y. 

2 1 .  Expand  {a^y^  —  3  xy)  ^ 

22.  If  a  man  can  ride  one  mile  for  a  cents,  how  far  can 
two  men  ride  for  b  cents  ? 

23.  In  how  many  days  can  x  men  earn  as  much  as  8 
men  in  y  days  ? 

24.  a  times  &  is  how  many  times  c? 

18.   Illus.     -3ab^\-6  o?W  +  15  a'b^  -  3  ab' 
20"    -    5ab   +    62 

To  divide  a  polynomial  by  a  inonomial,  divide  eaeh 
term  of  the  dividend  by  the  divisor,  and  add  the 
quotients. 

Exercise  25. 

Divide : 

1.  l%a'b^-A2o?b''  +  ^0a%x   by   ^o?b. 

2.  10 3.^  +  60^2/2- 18 a;y  by   2c(?y. 

3.  72ar*/-36icy-18a;y   by   ^x'y. 

4.  169  a^6- 117 a^^62_^  91^26   by   13 a^. 

5.  -2aV  +  |aV  by   fa^o;. 

6.  ^a:^y^-3x^y^   by    -fa;^2/'- 

7.  32«32/^2«-24a:52/V  +  8a;y  by    —Sa^y. 

8.  120 a^6V- 186 a^6^c«   by   6a^b^c\ 

9.  4a^-10aj^  +  2a;«-16aj3-6a;*   by   4a^. 
10.  242/3 +  32/ -48/  by   32/'. 


OPERA  riONS.  66 

11.  \a^x  —  ^  abx  —  focaj  by  lax. 

12.  -far»  +  fxy  +  Vicby  -|x. 

13.  An  army  was  drawn  up  with  x  men  in  front  and  y 
men  deep.     How  many  men  were  there  in  the  ai*my? 

14.  In  how  many  minutes  will  a  train  go  x  miles  at 
the  rate  of  a  miles  an  hour  ? 

15.  How  many  apples  at  x  cents  apiece  can  be  bought 
for  b  dollars  ? 

19.  Since  division  is  the  reverse  of  multiplication, 
let  us  consider  an  example  in  the  multiplication  of  poly- 
nomials. .     . 

a^  +  2x'  +  3x  r-fA^ 


3iB»-f-6aJ*H-9a?     f 

3a?-\-6x^-\-9x 


3ar'  +  4aj*-|-8a;3  -\-9x 

Which  of  these  numbers  will  become  the  dividend  in 
our  division?  Take  a^-\-2x^ -\-Sx  for  the  divisor.  What 
will  be  the  quotient?  Write  these  names  opposite  the 
different  numbers.  What  is  the  last  operation  performed 
in  obtaining  the  dividend?  What  then  is  the  dividend? 
How  are  these  partial  products  obtained  ? 

Keeping  these  facts  in  mind,  we  will  start  on  the 
work  of  division,  using  these  same  numbers  for  con- 
venience, 


66  A   FIRST  BOOK  IN  ALGEBRA. 


Illus.  1.* 

ar^  +  2a;2  +  3; 

x)3a^-^4.x' 

-hSx^  +  9x{3x^- 

3x'-{-6x' 

■i-da^ 

. 

-2x' 

-    x^-^9x 

-2a;4 

_4a;3_6ic2 

Sx'  +  ex^-i-dx 

Sx^-\-6x^  +  9x 

2a;  +  3 


How  was  the  first  term  of  the  dividend  formed  ?  How 
can  the  first  term  of  the  quotient  be  found  ?  Knowing  the 
divisor  and  first  term  of  the  quotient,  what  can  be  formed  ? 
Subtract  this  from  the  dividend.  How  was  the  first  term  of 
the  remainder  formed  ?  What  can  now  be  found  ?  How  ? 
What  can  then  be  formed?  etc.,  etc.  How  long  can  this 
process  be  continued? 

Illus.  2. 

^-3xy+2y')x'-6x^y-{-12x^f-Ay\a^-Sxy-\-f+  .^"^'f ~^^'  , 

x^—3xy+2y^ 

ar4_3a^2/+  2x'y'- 


■3x^y-\-10xy-iy* 
3a^2/4-  9x'y^-6xf 


a^y'^-^6xf-4:y* 
x~y^—3xy^-\-2y* 

9xy^—6y^ 

Why  stop  the  work  at  the  point  given  ?  What  is  the  com- 
plete quotient?  Is  the  dividend  exactly  divisible  by  the 
divisor  ?    When  is  one  number  exactly  divisible  by  another  ? 

*  It  would  be  well  for  the  teacher  to  work  out  this  example  on  the  board 
with  the  class  along  the  line  of  the  questions  which  follow  the  example. 


OPERATIONS.  67 

To  divide  a  polynomicd  hy  a  polynomiaZ,  arrange 
the  terms  of  the  dividend  and  divisor  similarly,  divide 
the  first  term  of  the  dividend  hy  tJie  first  term  of  tJie 
divisor  for  the  first  term  of  the  quotient,  multiply 
the  divisor  hy  the  quotient  and  suhtract  the  product 
from,  the  dividend ;  divide  the  first  term,  of  the  re- 
mainder hy  tlie  first  term  of  the  divisor  for  the  next 
term  of  the  quotient,  multiply  and  subtract  as  hefore; 
continue  this  work  of  dividing,  m^ultiplying,  and  sub- 
tracting until  there  is  no  remainder  or  until  the  first 
term  of  tJie  remainder  is  not  divisible  by  the  first  term 
of  the  divisor. 

Exercise  26. 
Divide : 

1.  a^  +  8a;  — 105by  a;-|-15. 

2.  «*  +  8a;-33by  a;-f  11. 

3.  a;*-|-x2_20by  x=-4. 

4.  y-y2_30byy«4-5. 

6.  a;*-31ar^-|-9by  iB2  +  5x-3. 

6.  rt* -120"  4- 16  by  a'^- 2a -4. 

7.  7?  —  tfhyx  —  y. 

8.  a»  4-  &*  by  a  -f  6. 

9.  16a* -816*  by  2a -36. 

10.  81a«-y*  by  3x^-2^. 

11.  7?-3^y-2xY-^3?f-\lxy'-12rf  by  x'-2xy-dy\ 

12.  a*  +  a*6-  13a»6'  +  ISa^d^  - 46«  by  a«  -  3a6  +  26^ 


68  A   FIRST  BOOK  IN  ALGEBRA. 

13.  x^  -  Bx;'  +  8  -\-  5x'  -  lOx  -  x^  -{-  lOx^  hj  x^ -^  8  -  x. 

14.  x''-2x'-2  +  x-8x'-\-2x'-5x'"hy  x'+'J-^x. 

15.  a'-a-2a--a^hj  a-\-a^-{-a\ 

16.  x^-2x^-x''-x'hyl  +  x'-i-x. 

17.  a"-a^by  a^  _  i. 

18.  a^^  —  a'^hj  a^  +  1. 

19.  a;^  +  42/*by  a;2-2aj^  +  22/2. 

20.  Aa*-{-Slb*hy2a'-\-6ab  +  9b\ 

21.  ia^^  +  f  0^2/  -  i^V  +  iW'  -  W  by  x'  -  ixy  +  f. 

23.  i(a;-^)^-(a;-2/)3_|(a;-2/)2-TV(^-2/)  by 

24.  16aj«  -  81?/^  by  272/'  +  18a^2/'  +  8a;H  12  A- 

25.  4a;^-10a^4-6by  a;  +  l. 

26.  4a*-5a262  +  54by  2a-&. 

27.  a^-i'^  +  a^  +  ft^  +  a^d^by  a2-62^1. 

28.  aj''-2/'-a^-?/^-a;y  by  ic2-2/2-l. 

29.  Sla''-16b^hyl2a^b*-8h'-lSa%^-\-27a^. 

30.  1  +  3aj  by  1  +  ic  to  four  terms  of  quotient. 

31.  1  —  2a  by  1  —  a  to  four  terms  of  quotient. 

32.  4  +  a  by  2  —  a  to  four  terms. 

33.  9  —  a?  by  3 -f  a;  to  four  terms. 

34.  If  a  boy  can  do  a  piece  of  work  in  x  minutes,  how 
many  hours  would  it  take  him  to  perform  12  times  as  much 
work? 


OPERA  TIONS.  69 

36.  A  man  has  x  dollars,  y  acres  of  land  worth  m  dollars 
an  acre,  and  c  houses  each  worth  b  dollars.  What  is  my 
share  if  I  am  one  of  n  heirs  ? 

36.  A  storekeeper  mixed  m  pounds  of  coffee  worth  a 
cents  a  pound  with  p  pounds  worth  b  cents  a  pound.  How 
much  is  the  mixture  worth  per  pound  ? 

37.  If  John  is  y  years  old,  how  old  was  he  11  years  ago  ? 

EVOLUTION. 

20.  Illus.  Va,    V^,    Vx  —  y^    VlC,    V5,    VOa^bc. 

The  root  of  a  number  is  indicated  by  the  radical 
sign  and  index.  When  no  index  is  expressed,  two  is 
understood. 

Express : 

1.  The  square  root  of  a?,     2ab,     7  a;  — 3?^,     a^bc. 

2.  The  fifth  root  oiSy,     2m-  n,     4  x^yz^. 

3.  The  cube  root  of  2,     x  +  y,     17  a^y*,    m. 

4.  The  sixth  root  of  az,     5  m-n  —  3  ic^/  +  14  —  3  abh. 

What  is  the  square  root  of  a  number?  the  fifth  root? 
the  fourth  root  ?  the  cube  root  ?  the  eleventh  root  ? 

Illus.  1.   (3a-6(r')3  =  ?  Then    ^/27a«6V=? 

Illus.  2.   (+a)^=:?>  ,^ 

(_ay=?[  ^^^'^    </T^*  =  ? 

(4-a)»=?  ^5^4^^  =  ? 

(-a)»  =  ?  </3^=? 


70  A   FIRST  BOOK  IN  ALGEBRA. 

To  find  the  root  of  a  monomial,  find  the  required 
root  of  the  coefficient,  divide  the  exponent  of  each 
letter  hy  the  indejc  of  the  root  for  the  exponent  of 
that  letter  in  the  root,  give  to  even  roots  of  plus  num- 
bers the  plus-or-minus  sign  (±),  to  odd  roots  of  plus 
numbers  the  plus  sign,  and  to  odd  roots  of  minus 
numbers  the  minus  sign. 

Exercise  27. 

Simplify : 

1.  Vl6^^.  9.    Vp^. 

2.  ^8l^.  10.    ^J, 


3.    ^-8a;y.  11.   -^_|Ja;Yl 


4.    V-32a^«6^.  12.    V-2%aW^ 


5.  V243a^6i«.  13.    </x'^{a-hY. 

6.  v^27^.  14.    -Va\x'+yy. 

7.  -v^ie^.  15.    ^^.a'Wix'-yy. 


17.    Vi"^-^|^^-A/^%"^^H-^^. 


18.  V|W  +  V  -  A^^'V  -  V^^^A"  -  Via;V 

19.  Multiply  V25a^6Vby  -  "vZ-Sa^ftV. 

20.  Divide  Vl44a;y  by  a/243 x^y. 


21.  Multiply  -  V-  n^x'y^^  by  V9icy;2«. 

22.  Divide  VlOOa^fe^^  ^y  ^32^15^5. 


OPERA  TIONS. 


71 


23.  From  two  cities  a  miles  apart  two  trains  start 
toward  each  other,  the  one  going  x  miles  an  hour,  and  the 
other  y  miles  an  hour.  How  long  before  they  will  meet? 
How  far  will  each  train  have  gone  ? 

24.  What  number  is  that  whose  double  exceeds  its  half 
by  39? 

25.  Mr.  A.  is  m  years  old,  and  6  years  ago  he  was  one- 
half  as  old  as  Mr.  B.  How  old  was  Mr.  B.  then  ?  How 
old  is  Mr.  B.  now? 


21.   (a  4-  hY  =  «'  -\-2ab  +  h-  =  «-'  +  (2a  +  h)h. 
Illus.  1.  Find  the  square  root  of  9a:^  —  V2xy\-  Ay^. 

3x-2y 


93^-12xy-\-4f 
9a^ 


2a-\-b  =  6x-2y 
(2a-^b)b  = 


-r2xy-\-4f 
^12xy-^4y^ 


Illus.  2.  Find  the  square  root  of  af'  —  6ar'  —  4a;-|-l+6a^ 


7fi-2x^-h5x*-Q3i^-\-Qx^-ix-i- 1 

3fi 


2x»-x2 


•2a*+   X* 


2(a?-x2)+2a5=2x»-2a;«+2x 


4x«-6a:«+6x2-4a:+l 
4x«-4x«+4x2 


2(x»-x2+2x)+(-l)=2x«-2xH4x  -  1 


■2x»+2x«-4x+l 
•2x«+2x2-4x+l. 


To  find  the  square  root  of  a  polynomial,  arrange  the 
terms  with  reference  to  the  powers  of  some  number; 
take  the  square  root  of  the  first  term  of  the  poly- 


12  A   FIRST  BOOK  IN  ALGEBRA. 

nomial  for  the  first  terin  of  the  root,  and  subtract  its 
square  from  the  polynomial ;  divide  the  first  term  of 
the  remainder  by  twice  the  root  found  for  the  next 
term  of  the  root,  and  add  the  quotient  to  the  trial 
divisor;  multiply  the  complete  divisor  by  the  second 
term  of  the  root,  and  subtract  the  product  from  the 
reinainder.  If  there  is  still  a  remainder,  consider  the 
root  already  found  as  one  term,,  and  proceed  as  before. 

Exercise  28. 

Find  the  square  root  of  each  of  the  following : 

1.  4.x'-12xy  +  ^y\ 

2.  a;^  +  10a^/-i-25a;y. 

3 .  16  a^bh"  -  ^Q  ahc'xyh  +  49  x'y^z^ 

4.  \x^  —  xy'^z-\-y^z^. 

5.  a26«  +  |a6«c^  +  i-c». 

6.  a;*  — 4 0.-^  + 2  a;- 4-4  a; +  1. 

7.  aj*  +  6ar^  + 170^2  + 24a; +  16. 

8.  4a;^  +  9a;2_|_4_4a,_4aj3 

9.  6a;»-5a^  +  l-2a;  +  9a5*. 

10.  x^-\-2of-x^  +  3x^-2x-\-l. 

11.  6a;4-4ar^  +  4a^-loa;2-8a;H-a;«H-16. 

12.  lx'-Qx-llx''  +  10x^-^x^^l  +  4.:xfi. 

13.  A  is  a;  years  old.     If  his  age  is  as  much  above  50  as 
B's  age  is  below  40,  what  is  B's  age  ? 

14.  If  a;  represents  the  first  digit,  and  y  the  second  digit, 
of  a  number,  what  will  represent  the  number  ? 


OPERA  TIONS.  73 

22.   (a-h6)'=a«-f3tt26  +  3a6*+6»=a«H-(3a'+3a6-h62)6. 
Illus.  1.     Find  the  cube  root  of  8m*  + 12mV  +  6mV 

8m«+ 12m*»H  6mV+ n»  [2nr-j-n^ 
a3=  8m« 


3a«H-3a6+6*=12m^+6mV4-n« 
(3a»4-3a6+6-)6= 


12mV+6m-M«+n» 


Illus.  2.     Find  the  cube  root  of  66a:*  +  33a^4-8  -  36a; 
4-ic*-63dr»-9x^. 


a<-9icS-|-33x*-63ag«+66a;«-36a;+8|a;g-3a;+2 
a* 


3x*-9x«+9a;2 


-0a*+33ar«-63a:«+66a;2-36a;+8 
-9x«+27a:*-27x» 


3(a;2  -  3x)2 = 3x*  -  18x« + 27x-* 
3(x2-3a;)x2=  6a;2- 18x 

2-^=  +4 


3ar«-18a:«+33a:2_i8x+4 


6x*-36a^+66a;2-36a;+8 


6ie4_36a:8+66a:2-36x+8 


To  y^n^  ^/le  cz<'&e  root  of  a  polynomial,  arrange 
the  terms  with  reference  to  the  powers  of  some  num- 
ber; take  the  cube  root  of  the  first  term^  for  tli^  first 
term  of  the  root,  and  subtrax^t  its  cube  from  the 
polynomial;  take  three  tim^s  the  square  of  the  root 
already  found  for  a  trial  divisor,  divide  the  first 
term  of  the  remainder  by  it,  and  write  the  quotient 
for  the  next  term  of  tJie  root ;  add  to  tlie  trial 
divisor  three  times  the  product  of  the  first  term  by 
the  second  and  the  square  of  the  second-;  multiply 


74  A   FIRST  BOOK  IN  ALGEBRA. 

the  complete  divisor  hy  the  second  term  of  the  root, 
and  subtract  the  product  from  the  remainder.  If 
there  are  other  terms  remaining,  consider  the  root 
already  found  as  one  term,  and  proceed  as  before. 

Exercise  29. 

Find  the  cube  root  of  each  of  the  following ; 

1.  27a^-27a^2/4- 9a;/-2/«. 

2.  15ar^-l-75a;4  +  125a^. 

3.  144a262  +  276<'+108a&*  +  64al 

4.  x^-Sy^-\-12x'y^-e>xY. 

5.  l-\-9x  +  21o^-\-21x\ 

6.  1  -  21  m  -  343  m^H- 147  m^. 

7.  3ic2_|_g4^6_24a;4_  j_ 

8.  27a^^+2T  +  «^  +  9a;«. 

9.  8ft«  +  18a^  +  9a2-12a^-13a3-3rt  +  l. 

10.  a;9-7ic«  +  ar^-3a;^-3a^4-6a;^  +  6a^. 

11.  5ic3-3a;-3a^-l  +  aj^. 

12.  i-a«  +  f  a^  +  ^a""  +  2a^-  lOJa^  -f  6a  -  1. 

13.  A  board   is   8a;   inches  long   and   2x  inches  wide. 
What  is  the  length  of  a  square  board  having  the  same  area  ? 

14.  If  3a;  represents  the  edge  of   a  cubical  box,  what 
represents  the  cubical  contents  of  the  box  ? 


OPERATIONS.  75 

15.  If  a  cubical  cistern  contains  642/*  cubic  feet,  how 
long  is  one  edge  ? 

16.  A  man  travels  north  a  miles,  and  then  south  b  miles. 
How  far  is  he  from  the  starting-point?  How  far  has  he 
traveled  ? 

Kxercise  30. 

Find  the  indicated  roots : 


1.  V2025.  5.   V70.56.  9.    V59.319. 

2.  V9409.  6.    V:9025.  10.   VSSdOlT. 


3.    V20449.  7.    V94864.  11.    V2418()4.367. 


4.   V904401.         8.   V.00000784.        12.   V7039.21. 
13.   V36.287552.  14.   V2550.25. 


15.  V34. 78312   to  three  decimal  places. 

16.  V7  to  three  decimal  places. 

17.  V.1255  to  three  decimal  places. 

18.  A  merchant  bought  a  bale  of  cloth  containing  just 
as  many  pieces  as  there  were  yards  in  each  piece.  The 
whole  number  of  yards  was  1089.  What  was  the  number 
of  pieces  ? 

19.  A  regiment,  consisting  of  5476  men,  is  to  be  formed 
into  a  solid  square.  'How  many  men  must  be  placed  in 
each  rank  Z 

20.  /What  is  the  depth  of  a  cubical  cistern  which  contains 
5000  gallons  of  water  ?     (1  gallon  =  231  cubic  inches.) 

21.  A  farmer  plants  an  orchard  containing  8464  trees, 
and  has  as  many  rows  of  trees  as  there  are  trees  in  each 
row.     What  is  the  number  of  trees  in  each  row  ? 


FACTORS   AND   MULTIPLES. 


FACTORING. 

23.  When  we  divide  a  number  into  two  numbers, 
which  multiplied  together  will  give  a  product  equal  to 
the  given  number,  we  have  found  the  factors  of  that 
number.     This  process  is  called  factoring. 

Name  two  factors  of  48,  27,  18,  35,  49,  72. 
Name  three  factors  of  24,  100,  75,  64,  72,  40. 

24.  CASE  I.  To  factor  a  polynomial  which  has 
a  factor  common  to  all  its  terms. 

Illus.  Factor  2ab  -^6aG -\-4:ad. 

6  +  3c    -{-2d 
.-.  2ab-\-6ac  +  4:ad  =  2a(b-\-3c-\-2d). 

Divide  the  polynomial  by  the  largest  factor  com- 
mon to  the  terms.  The  quotient  and  divisor  are  the 
factors  of  the  polynor/bial. 

n 


FACTORS  AND  MULTIPLES.  77 

Exercise  31. 

Factor  : 

1.  5a2-25.  7.  a^-od*. 

2.  16-|-64xy.  8.  a-  +  ah. 

3.  2a-2a\  9.  6a'4-2a^  +  4a^ 

4.  15a2-225a*.  10.  7a5- 7a:3+ 14x*. 

5.  a?-a^.  11.  3a^-a?2  +  a;. 

6.  3 a* -ha*.  12.  a^  —  a-y  +  ay'. 
13.  3a(«  +  2/)  +  5m6(a;  +  y)  — 9d=^x(a;-f  y). 

X  14.   4(a  -  6)  -  15x2/(a  _  6)  +  (a  -  6)  -  5a'b{a  -  6). 
16.   4x»y-12jB2y2_3a.^ 

16.  6aa^f-4ax^f  +  2axy''-2a^xf. 

17.  51  a:*y- 34 xy  + 17 ar^y. 

18.  6a*62-3a«6»c-9a6»c  +  3a6c2. 

19.  3ax^  — 24 ox +  9 aar^  — 3 ax*  — 9 aa^. 

20.  27  a«6V  -  81  a^ftV  +  81  a«6V  -  27  a*6V  -  27  a'b^c\ 

21.  A  lady  bought  a  ribbon  for  m  cents,  some  tape  for  d 
cents,  and  some  thread  for  c  cents,  for  which  she  paid  x 
cents  on  account.     How  much  remains  due? 

22.  Julia  has  the  same  number  of  beads  in  each  hand. 
If  she  should  change  two  from  her  left  hand  to  her  right 
hand,  the  right  hand  would  contain  twice  as  many  as  the 
left.     How  many  beads  has  she? 

Sometimes  a  polynomial  in  the  form  given  has  no 
factor  common  to  all  its  terms,  but  has  factors  com- 
mon to  sevei-al  terms.     It  may  be  possible  then  to 


78  A    FIRST  BOOK  IN  ALGEBRA. 

group  the  terms  in  parentheses  so  that  there  will  be 
a  binomial  or  trinomial  factor  common  to  all  the  terms 
of  the  polynomial  in  its  new  form. 

Illus.  1.  Factor  2  am  -i- 2 ax  -\- bm  -\-  hx. 

2am -\-  2ax  -\-bm  -\-  bx  =  2a{m  -\-x)  -{-  b{m-{-x). 

m-\-x  \  2a{m  -\-x)  -\-b{m-\-  x) 
"2a  -\-b 

.'.  2am-\-2  ax  -{-bm-\-bx  =  (m  -{-x){2a-\-  b). 

Illus.  2.  Factor  a*  +  a^  —  a^  —  a^  -\-a^-\-l. 

a'  +  a'-  a'  -  a"  +  a^  +  1  =  a^a""  -f- 1)  -  a\a^  +  1)4-  (a^  +  1). 

a^  -f  1  I  a^{a^  +  1)  -  a^a"  +  1)  +  {a^  +  1) 
a«  -a^  +1 

...  a«  +  «'  -  a^  -  a'  +  a^  +  1  =  (a^  + 1)  (a«  -^  a^  + 1) . 

Exercise  32. 

Factor : 

1.  ax  +  ay  +  bx-\-by.  6.  y? -{-  mxy  —  4 x?/  —  4 my'''. 

2.  x^  +  ax -[- bx  ^  ab.  7.  2a;^  — a^  +  4a;  — 2. 

3.  ax^ -\- ay^  —  bxr  —  by^.  8.  mx  —  ma  —  nx -\- na. 

4.  x^'  —  ax-\-5x  —  5a.  9.  a^ -{- x^  -{- x -\- 1. 

5.  aa?  —  &a;  4- a6  —  a;l  10.  if  —  y^-^-y  —  l, 

11.  a:^  +  a;'^  —  cc'*^  —  a;2  4-  ic  4- 1. 

12.  a^a;  4-  ctbx  4\  ac)4-toy)4-'\&^2/l  +M 

13.  ax  —  bx -\- by  -{- cy  —  ex  —  a?/.  \ 

14.  3aa;4-3a?/  — 2&X  — 26^. 

15.  2ax  —  Sby-{-cy  —  2ay-\-Sbx  —  cx. 

16.  6ama;4- 3am2/  — Gawaj  — 3an2/. 


FACTORS  AND  MULTIPLES.  79 

17.  A  man  had  250  acres  of  land  and  30  houses.  After 
trading  x  of  his  houses  for  y  acres  of  land,  how  many  has 
he  of  each  ? 

18.  What  is  the  number  that  will  become  four  times 
as  large  by  adding -36  to  it? 

19.  The  fifth  and  seventh  of  a  number  are  together 
equal  to  24.     What  is  the  number? 

25.  CASE  II.  To  factor  the  difference  of  the 
squares  of  two  numbers. 

iLLUS.  1.  a^  -  62  =  (a  +  b)(a-b). 

Illus.  2.  9  x'y*  -  49  a^d'^  ^  (3 ^y2  _^  7  ^2^6)  {Sxy^-7  a^b^) . 

Illus.  3. 

a^-y'  =  {ai'  +  y')ix'-^n  =  i^'  +  y*)(^  +  f)i^-y') 
=  {^  +  ?/)  {a^  -f  y^)  {x  +  y){x-y). 

Write  two  factors,  of  which  one  is  the  sum  and 
the  other  tlie  difference  of  the  square  roots  of  the 
terms. 

Exercise  33. 

Factor: 

1.  7? -'if,  7.   shf-ay. 

2.  m^  —  v}.  8.  •  gf*c*  —  «*2*. 

3.  a^W-(?d?.  9.   4a2-9a?2. 

4.  mY-7^y^.  10.    16m'*-9n*. 

5.  a%''7^-m'di'\  11.   81iry-256*d^. 

6.  a:y«* -c2(fm\  12.    729mVa;^«- 10,000 y^ 


80  A    FIRST  BOOK   IN  ALGEBRA. 

13.  121m^-64.x\  19.  a^  -  ax^ 

14.  x'-y\  20.  5b'-5a^b\ 

15.  m^  —  a\  21 .  ax^  —  ay^  —hx-  +  by-. 

16.  a%^  —  1.  22.  5ax  —  ^a^x-\-^ay  —  b a^y. 

17.  aj^^  — &^^  23.  m^  —  y^  —  am -\- ay. 

18.  16a^-l.  ^4.  a2_^2_^_^_ 

25.  By  how  much  does  x  exceed  Sy? 

26.  Eleven  years  ago  C  was  three  times  as  old  ac  D 
whose  age  was  x  years.     What  is  C's  present  age  ? 

26.  CASE  III.  To  factor  the  sum  of  the  cuhes 
of  two  numbers. 

Illus.  1.     Divide   a^  +  6^  by  a  +  6. 

Illus.  2.     Divide   m^  -|-  ny  by  m  +  ny. 

Illus.  3.     Divide   Sa^^  +  6V  by  2  a-  +  bc^ 

Notice  in  each  case  above  : 

1.  That  the  divisor  is  the  sum  of  the  cube  roots  of  the 
terms  of  the  dividend. 

2.  That  the  quotient  is  the  square  of  the  first  term  of 
the  divisor,  minus  the  product  of  the  first  term  by  the 
second,  plus  the  square  of  the  second. 

Write  two  factors,  one  of  which  is  the  sum  of  the 
cube  roots  of  the  terms,  and  the  other  the  quotient 
obtained  by  dividing  the  original  number  by  the 
first  factor. 


FACTORS  AND  MULTIPLES.  81 

Exercise  34. 

Factor : 

1.  a^4-y'.  12.  1-I-6V. 

2.  <r»-fd».  13.  ^a^'  +  d". 

3.  a'  +  6V.  14.  ^iB3  +  l. 

4.  aV-h!/^.  15.  (m4-w)*H-8. 

5.  8a«6V-hwi«.  16.  l  +  (a;-y)' 

6.  x«y'  +  216a^  17.  2aV/+2a«6». 

7 .  a®  +  6^  18.  ?»x*  +  my  —  7i.T  —  )iy. 

8.  64 3:^4-/.  19.  a?x -\- b^x  —  ahj  —  bhj. 

9.  0^4-8.  20.  a«-6« 

10.  21-\-a^h\  21.   729iB«-642/^ 

11.  y^-fl.  22.   l-o". 

23.  A  had  d  dollars,  but,  after  giving  $26  to  B,  he  had 
one-third  as  many  as  13.  How  many  has  B  ?  How  many 
had  Bat  first? 

24.  How  many  units  in  y  tens  ? 

26.  If  the  sum  of  two  numbers  is  a;,  and  one  of  the 
numbers  is  8,  what  is  the  other  number  ? 

What  does  a'  mean  ?  What  does  3  a  stand  for  ?  What 
is  the  product  of  x^  and  0  ? 

27.  CASE  IV.  To  factor  the  difference  of  tlie 
cubes  of  two  numbers. 

Illus.  1.     Divide  a^  —  Why  a  —  b. 

Ili^vs.  2.    Divide  27  aV  -  c«d"  by  3  a?l^  -  cdi,\ 


82  A    FIRST  BOOK  IN  ALGEBRA. 

Notice  in  each  case  above : 

1.  That  the  divisor  is  the  difference  of  the  cube  roots  of 
the  terms  of  the  dividend. 

2.  That  the  quotient  is  the  square  of  the  first  term  of 
the  divisor,  plus  the  product  of  the  first  by  the  second,  plus 
the  square  of  the  second. 

Write  two  factors,  one  of  which  is  the  difference 
of  the  cube  roots  of  the  tej^ms,  and  the  other  the 
quotient  obtained  by  dividing  the  original  number 
by  the  first  factor. 

Exercise  35. 

Factor : 

1.  :(?-a\  13.  3?-l. 

2.  c^-W.  14.  1-f. 

3.  o?-xY'  15.  i^a?y^-h\ 

4.  mhi^-&.  16.  8-J^mV. 

5.  21m^-%a?.  17.  l-(a  +  6)3. 

6.  8ar^-64^.  18.  a^f-{x-yy. 

7.  64aV63-125mVv«.  19.  x' -  xy\ 

8.  27b^cy-216aVx^  20.  ab^  -  a(f  +  mb^  -  mc^. 

9.  8a;y  —  125.  21.  x^  —  /  into  four  factors. 
10.  27-64mVv^.  22.  x^ -x^  +  a^  — x^-x-^1. 


11.  a^-b^  23.    a'-b'  +  a'-b\ 

12.  m^  —  a^.  24.    ma? -^  ni'if  —  x  —  y. 


FACTORS  AND  MULTIPLES.  88 

25.  How  many  liours  will  it  take  x  men  to  dig  75 
bushels  of  |K)tatoe8  if  each  man  digs  y  bushels  an  hour  ? 

26.  If  there  are  x  tens,  y  units,  and  z  hundreds  in  a 
number,  what  will  represent  the  whole  number  of  units  ? 

28.  CASE  V.  To  factor  trinomials  which  are 
perfect  squares. 

Square  c  +  b^  c  — 6,  ^—y\  Smn^+y\  2a^bc  —  Sx^i/2^, 
How  are  the  first  and  last  terms  of  these  trinomial 
squares  formed?     How  is  the  middle  term  formed? 

When  is  a  trinomial  a  square  ? 

Name  those  of  the  following  trinomials  which  are 
squares : 

1.  m^-2mp+p\  5.  a*-lSa^  +  9. 

2.  x'-\-2xy-f.  6.  96*4-12^2  +  4. 

3.  ix'-ixy  +  f.  7.  iey*-Sy'-\-l. 

4.  x^-^-Gx'y  +  df.  8.  25€»d^-\- 20 c*d^x-\-Aa^. 

Illus.  1.     a*  H-  2a6  -f  6^  =  (a  +  &)  (a-{-b)  =  (a  +  by. 
Illus.  2.     a  -2ab  +  b'^  =  (a-  b)  (a  -  6)  =  (a  -  by. 

Illus.  3.     9a^ -12ab  +  4b^  =  (Sa-2by. 

I 

Write  two  hinomial  factors,  each  of  which  consists 
of  the  square  roots  of  the  squares  connected  hy  the 
si^n  of  the  middle  term. 


84  A    FIRST  BOOK  IN  ALGEBRA. 

Exercise  36. 

Factor : 

1.  a^  +  2ax  +  x'.  13.  9 c" -{- 66 cd  +  121  (P. 

2.  c2-2cd  +  dl  14.  4/ -362/ +  81. 

3.  4a2  +  4a?/  +  2/'«  15-  x"  -  6 xry -{- 9 y\ 

4.  a2  4-4a6  +  46l  16.  9  -  12a;'-^  + 4a;*. 

5.  x'-6cx-^9c\  17.  16a^2/'-24a^a;2/2  +  9a^ 

6.  16x^-Sxy-^y\  18.  25 6^  +  30 ic^?/ +  9 cV . 

7.  a2  +  2a  +  l.  19.  49 a^  +  28 aa:^?/ +  4 xy . 

8.  9-6a;  +  a;2^  20.  ^x'-^xY  +  y\ 

9.  a:^- 10a; +  25.  21.  \a^  +  2ab-{-U\ 

10.  4/ -12?/ +  9.  22.    lajy-fx^^/^^  +  z^ 

11.  9a^  +  24a;+16.  "^  23.    a;^  -  4a;y +  4a;3/. 

12.  81a^-18a2  +  l.  24.    18 a^^/ +  24 a/ +  8 /. 

25 .  3  aV  -  30  a'bx'  +  75  aft^a^. 

26.  (x-\-yy-2a\x  +  y)-^a\ 

V  27.  (m2-rt2)2_2(m*-n*)  +  (m2  +  w2)2. 

28.  a^  +  2ab-\-b'  +  6a-\-6b. 

29.  a,-2  +  2/'-3a;-2a;?/  +  32/. 

30.  x^-\-y''-2o?f. 

What  must  be  added  to  the  following  to  make  them 
perfect  squares  ? 


31.  x'  +  2xy.  33.    c^^d\ 

32.  m^  +  6mY  34.    {x  —  yy  +  2{x  —  y) 


2 


FACTORS  AND  MULTIPLES.  85 

35.  (c-rf)2-G(c-d).  37.    S0xy*  +  9a^f. 

36.  9a^^-12ic2y3  gg.   25  a^b" +  S6b*(^. 

39.  49a26V  +  25a26*rf^. 

40.  a*  —  22a*  4-  9  (two  numbers). 

41.  64a»-177ay  +  121/.  43.    a^-fa:*4-l. 

42.  a^-10a262-h96^  44.    a^  +  a  +  l. 

45.  A  stream  flows  at  the  rate  of  a  miles  an  hour,  and  a 
man  can  row  in  still  water  b  miles  an  hour.  How  far  can 
the  man  row  up  the  stream  in  an  hour  ?  In  6  hours  ?  How 
far  down  the  stream  in  an  hour  ?     In  3  hours  ? 

46.  A  cistern  can  be  filled  by  two  pipes  in  3  hours  and  5 
hours  respectively.  What  part  of  the  cistern  will  be  tilled 
by  both  pipes  running  together  for  one  hour  ? 

47.  Nine  years  ago  Henry  was  three  times  as  old  as 
Julius.  If  Henry  is  b  years  old,  how  old  was  Julius  then  ? 
How  old  is  Julius  now  ? 

29.   CASE  VI.      To  factor  trinomials  in  the  form 

x^±cx±  d. 

This  is  the  reverse  of  the  case  under  Multiplication 
given  in  Art.  14. 

Illus.  1.  ar*  -f  14a;  +  45  =  (a;  +  9)  (a;  -f  5). 

Illus.  2.  ar^  — 6a;  +  5  =  (a;  — 5)(a;  — 1). 

Illus.  3.  a*  +  2x  -  3  =  (a;  -f  3)  (a;  -  1). 

Illus.  4.  ar^  - 8x  -  20  =  (a;  -  10)  (x  +  2). 


86  A    FIRST  BOOK  IN  ALGEBRA. 

Write  two  binomial  factors,  the  first  term  of  each 
being  the  square  root  of  the  first  term  of  the  given 
trinomial,  and  for  the  second  terms  of  the  factors 
find  two  numbers  whose  algebraic  sum  is  the  coeffi- 
cient of  the  second  terin  and  whose  product  is  the 
last  term. 

Exercise  37. 

Factor : 

1.  a- 4- 3a +2.             *  15.  x^  +  A.x'-ll. 

2.  a;2  +  9a;  +  18.  16.  SO  +  Haj  +  x^. 

3.  a;2-5aj  +  6.  17.  21  +  10a  +  a\ 

4.  a^-7a  +  10.  18.  35  -  12rc  +  a;2. 

5.  /-IO2/  +  I6.  19.  36 -13a; +  0^2^ 

6.  C--C-6.            ^  20.  c2  +  2ccZ-3d2. 

7.  x^-\-4.x--6.  21.  a^  +  ^ax^l^x". 

8.  a;2-f.5aj-6.  22.  x^-xy-2^y\ 

9.  2/^ +  82/ -65.  23.  x'  +  x^y-12y\ 

10.  a^  — 4a  — 77.  24.  a;*  — 14a;y  +  452/*. 

11.  a;2-2a;-63.  25.  a,-^  -  23a;*  +  132a;3. 

12.  a2  +  10a-75.  26.  a«-12a^  +  35a^ 

13.  a2- 24a +  143.  27.  3aa;*  -  39aa;2  + 108a. 

14.  a;«-4a;3-117.  28.  Sa^- 12a26  +  12a6l 

29.  &d^  -  c^  -  ah^d?  +  a\ 

30.  ax^  -\-hx'^  —  5hx  —  6ax-\-Q>h  -\-Qa. 


FACTORS  AND  MULTIPLES.  87 

31.  A  field  can  be  mowed  by  two  men  in  a  hours  and 
b  hours  respectively.  What  part  of  the  field  can  be  mowed 
by  both  men  working  together  for  one  hour  ? 

32.  In  how  many  days  can  two  men  do  as  much  as  x 
men  in  7  days  ? 

Exercise  38.     (Review.) 

1.  Divide  50a  +  9a*  -h  24  -  67 a'  by  a  -f  a-  -  6. 

2.  Find  the  square  root  of 

3.  Expand  (a;  +  y)(aJ-i/)(ar^4-y'). 

4.  What  must  be  subtracted  from  a^  — 5a'^  +  27a  —  1 
to  produce  a  -h  1  ? 

5.  Expand  (sc^y-iz^ay. 

Factor : 

6.  a:2_iia.^l0.  n.    l-^-64a^. 

7.  (a  +  by-{c  +  dy.  12.    a«-l. 

8.  Ga;»-f 4a^-9a;-6.  13.    a^  +  2x--x-2. 

9.  9a^-.66iC«H-121ic*.  14.    a^-y^. 

10.  8c«-d».                              15.   27  +  12a;  +  a^. 

16.  Find  the  cube  root  of  96  m  —  64  —  40  m*  +  m^-\-6m\ 

17.  Simplify  (mH-2n)*- (2m- n)*. 

18.  Simplify  (5+7a;)  (5-7x)-(3+2a;)«+  (a;-9)  (x-i). 

19.  Simplify  (y  +  x)(f--xy  +  a^)-(y-x)(y'+yx+x'). 


88  A   FIRST  BOOK  IN  ALGEBRA. 

20.  Find  three  consecutive  numbers  whose  sum  is  57. 

21.  What  number,   being   increased   by   three-fifths   of 
itself,  will  equal  twice  itself  diminished  by  24  ? 

22.  Find  the  value  of  (4a+56)  (3a-f46) -a^ft+feV-c^ 
when  a  =  0,  5  =  1,  and  c  =  2. 


5^<K< 


GREATEST   COMMON   FACTOR. 

30.  What  are  the  factors  of  9  a^b^ +9a%^?  Of  3a*b-3a'b^? 
Which  are  factors  of  each  of  them  ?  What  is  the  largest 
number  that  is  a  factor  of  each  of  them?  This  number 
is  called  their  Greatest  Common  Factor.  What  factors  of 
the  numbers  does  it  contain? 

Illus.     Find  the  G.  C.  F.  of 

3ac  +  36cand  6 a^x-{- 12 abx  + 6 b^x. 

Sac+Sbc  =  3c(a  +  b) 
6a'x  +  12abx  -\-  (Jb'x=  6x  (a  -\-  b)  (a  +  b) 

G.C.F.  =3  (a +  5)  =  3a  4- 36 

To  find  the  G.  C.  F.  of  two  or  more  numbers,  find 
the  prime  factors  of  each  of  the  numbers  and  take 
the  product  of  the  com^mon  factors. 


FACTORS  AND  MULTIPLES.  89 

Exercise  30. 

Find  the  G.  C.  F.  of  each  of  the  following ; 

1.  3a*63  4-6a«6*  and   %a^W  +  l^ab\ 

2.  ^7?y  —  loQt?y^  and   15ary  — 45a^^ 

3.  2ir*3/*  +  2ar^3/'  and   Ga^f-Gxy". 

4.  3a«6-3a'6*  and   12  a^6*  -  12  a*6«. 

5.  mx  —  7na  —nx-\-  na   and   m^  —  n^ 

6.  81a!«-16  and   81  x» -  72 x*  + 16. 

7.  a^^20-x,   3r^-24a:  +  4o,   and   ar-5x. 
.    8.  a2  +  a;-6,   2^^-15-20;,   and  3ar^-27. 

9.    a* -16:   a^-a-e,   and    {a^-^y, 

10.  3/^-2^,   2/^-h92/2_10y,    and   y*-y. 

11.  ar  —  y^,   xy  —  y- -\- xz  —  yz,   and   x^  —  sc^y -\- xy^  —  y^. 

12.  3ac*cr-3oc2-3aVW  +  3a-^  and   9aV-9a«. 

13.  64a;"-8ic2/  and   12a:«4-3x22/^- 12a.V. 

14.  A  merchant  mixes  a  pounds  of  tea  worth  x  cents 
a  pound  with  b  pounds  worth  y  cents  a  pound.  How  much 
is  the  mixture  worth  per  pound? 

15.  If  a  man  bought  a  horse  for  x  dollars  and  sold  him 
so  as  to  gain  5%,  what  will  represent  the  number  of 
dollars  he  gained  ? 

16.  The  difference  between  two  numbers  is  6,  and  if  4 
be  added  to  the  greater,  the  result  will  be  three  times  the 
smaller.     What  are  the  numbers  ? 


90  A    FIRST  BOOK  IN  ALGEBRA. 

Of  how  many  terms  does  the  expression  sc^ —  Aoc^y  -\-y^ 
consist  ?  How  many  factors  has  each  of  the  terms  ? 
What  is  the  value  of  a  number,  one  of  whose  factors  is 
zero? 

LEAST    COMMON    MULTIPLE. 

31.  Illus.  1.  12  a^b^  is  the  least  common  multiple  of 
8a-b  and  4a&l  How  many  of  the  factors  of  Sa^b  are  found 
in  the  L.  C.  M.  ?     How  many  of  the  factors  of  4a6^? 

Illus.  2.  Find  the  L.  C.  M.  of  9ac  +  9bc,  3a^x-3b-x, 
and  ax^  —  bx^. 

9ac  +  9bc==9c(a  +  b) 
Sa'x-S b'x  =  Sx(a-\-  b)  (a  -  b) 
ax^  —  b3?  =  x^{a  —  b) 

L.  C.  M.  =  9ca^(a  +  b){a-  b) 

To  find  the  L.  C.  M.  of  two  or  more  numhers,  find 
the  prime  factors  of  each  number,  take  the  product 
of  the  different  factors,  using  each  the  greatest  num- 
ber of  times  it  is  found  in  any  of  the  numhers. 

Exercise  40. 

Find  the  L.  C.  M.  of  each  of  the  following: 

1.  a2-4  and   a^  +  Sa- 10. 

2.  0^  +  1   and   x'^-\-2x  +  l. 

3.  a^_2a;-15   and   x'-9. 

4.  a;*-4a^  +  3   and   x^  —  1. 


FACTORS  AND  MULTIPLES.  91 

5.  ar^-l,   iB2-4x  +  3,   and   2x'-2x-\2, 

6.  a3  +  l4-3a  +  3a^  and   am  —  2  —  2a-\-m. 

7.  a2-4,   a2  +  3a-10,   0^-25,    a--9,   a^-Sa  +  lo, 
and  a^  -h  5  a  +  6. 

8.  ar»-l-3a:2_|_3a.  aujj  a^_3_y _|_3a.. 

9.  ar'-l,     a:2_9^     a2-2a;-3,     x^-16,     a?-x-\2, 

10.  a:«  +  a^,   2a;*-2ic2,   and   a?-[-x. 

11.  a^  +  2«2  +  i,   l_2a2  +  a*,   and   1-a*. 

12.  ah  —  srz,  Sox  —  Sa^j   and  2ax  +  2aK 

13.  a?M  +  «>i  — 3m  —  3?i,  ?>i^  — 7r,  and  a-  — 10a +  21. 

14.  36-362,  l_6»,  and2a;-26*a;. 

15.  (I'—ncr—a—Xj  5a*6— 10a6»+56aj*,  and  a^b—abx—ab. 

16.  Tlie  thei-mometer  now  indicates  c  degrees  above  zero, 
but  yesterday  it  indicated  y  degrees  below  zero.  How  much 
warmer  is  it  to-day  than  yesterday  ? 

17.  A  certain  county  extends  from  b  degrees  north  lati- 
tude to  y  degrees  north  latitude.  What  is  the  extent  of 
the  county  from  north  to  south  ? 

18.  If  A  can  build  a  wall  in  x  days,  what  part  of  the 
wall  will  he  build  in  two  days  ? 

19.  How  many  times  can  a^-fl2  be  subtracted  from 
x«-f  24a:»  +  144? 

20.  Divide  $2142  between  two  men  so  that  one  shall 
receive  six  times  as  much  as  the  other. 


92  A    FIRST  BOOK   TN   ALGEBRA. 

Exercise  41.* 

1.  Reduce  to  lowest  terms  : 

6"?   9"?  TTT'  T¥'  T¥?  TS"?  "22 J  YT?  "8>  A'  T^'IJ'J  f eU"?  "STT* 

2.  Change : 

a.  i  to  8ths.  d.    f  to  25tlis.  g.    f  to  28ths. 

b.  f  to  12tlis.  e.     2  to  21sts.  h.    |  to  36ths. 

c.  I  to  32ds.  /.    3-9^  to  50ths.  i.    -^%  to  39ths. 

3.  How  many  15ths  in  f,  |,  f,  3,  If,  2|? 

4.  How  many  12ths  in  i  |,  |,  4,  9,  2i  7f  ? 

5.  Change  to  equivalent  fractions  : 

a.    2f  c.    12|.  e.    9f 

6.    3f.  d.    27f  /.    18f. 

6.  Change  to  equivalent  entire  or  mixed  numbers : 


a. 

¥• 

d.    ff. 

3-    ^F- 

b. 

¥• 

e-    ¥• 

h.  H- 

c. 

¥• 

/■   ¥■ 

J.     Ifi. 

7.    Change  to  equivalent  fractions 

having  L.  C.  D. : 

a. 

h  h  i- 

«•    i.ixV 

^.        hhh 

b. 

8.   Add: 

d-  l,i,i- 

f'       \^.hhh\' 

a. 

1  and  4. 

d.    1  and  |. 

g,    2iand3i. 

b. 

i  and  f 

e.    fandf 

h.    42  and  11^-. 

c. 

i  and  i. 

/    fandf 

i.     14|  and  27f . 

*  This  exercise  should  be  conducted  orally,  and  if,  at  any  point,  the 
students  do  not  readily  recall  the  method,  the  examples  of  that  class 
should  be  duplicated  till  the  principle  is  clear. 


FACTORS  AND   MULTIPLES.  93 


9.    Subtract: 

a.    \  from  \. 

d.    JfromTJj. 

9- 

2i  from  5f 

b.    1  from  \. 

e.    ^  from  |. 

h. 

6f  from  15^. 

c.    4  from  J. 

/.    f  from  J. 

i. 

27J  from  29i. 

10.    Find  the  product  of : 

. 

a.    ^by3. 

g.   2iby5. 

m. 

MbyH. 

b.    A  by  2. 

ft.    12|by3. 

n. 

15Jby|. 

c.    fbyG. 

i.    25by2f. 

0, 

24i  by  f . 

d.    5  by  J. 

J-  ibyi- 

P- 

36ibyf 

6.     9byT7y. 

k.    ibyf 

Q- 

2i  by  5i. 

/    4byf 

i.     fbyf 

r. 

3f  by  4|. 

11.    Divide: 

a.    i^byS. 

J.    679iby6. 

m. 

«byA- 

b.    i^by5. 

h.    ibyi. 

n. 

SJbylJ. 

c.    ^  by  4. 

i.    ibyf 

0. 

3fby2J. 

d.    ibyl2. 

J-    4by|. 

P- 

48  by  3i. 

e.    728Hby7. 

fe.    6by|. 

g- 

63  by  4J. 

/.    843iby8. 

'•    9by|.    , 

r. 

fbyf. 

12.    Simplify: 

a-   J+^- 

■i  =  ? 

c.  4- 

-i+J=? 

6.    *  +  «  = 

:  ? 

d.  ?i 

U? 

}-J 

4^ 

13.  16  is  ^  of  what  number  ? 

14.  }  is  what  part  of  7  ? 

15.  J  is  what  part  of  |  ? 

16.  What  is  I  of -^  ? 

17.  If  I  of  a  number  is  20,  what  is  J  of  the  number  ? 

18.  I  of  60  is  I  of  what  number  ? 


FRACTIONS. 


32.  In  the  previous  exercise  the  different  operations 
performed  upon  fractions  in  arithmetic  have  been  re- 
viewed. The  principles  and  methods  of  operation  are 
the  same  in  algebra. 

REDUCTION   OF  FRACTIONS. 

33.  Illus.  1.  Keduce  -- — -  to  lowest  terms. 

10  ab^ 

5a^b  -^  Bab        a 


10ab~^5ab      26 

d  til* Jt^ 

Illus.  2.  Reduce  — to  lowest  terms. 

a^bx  —  ab'-x 
• 
a^bx  —  b^x  _  bx{a^  —  b^)-7-  bx  {a  —  b)_  a  -\-b 

arbx  —  ab^x     abx  {a  —  b)-i-  bx  (a  —  b)         a 

To  reduce  a  fraction  to  its  lowest  terms,  divide  the 
terms  of  the  fraction  hy  their  greatest  common  factor. 

Exercise  42. 

Reduce  to  lowest  terms : 


2     12  ar'yV  ^     a;^  +  9a;  +  18 

mxf^'  '    x'-2x-W 

94 


FRACTIONS.  95 


6.    ^^-^. 


mx  —  ny -\- iix  —  my  ^^ 

ax  —  2hy-\'2hx  —  ay  '     ax^  —  x^y 


ac 

-bd  +  ad- 

be 

ax  — 

2hy  +  2ay 

-bx 

ac'd 

-a'^d^ 

a'c- 

f  a*d 

a-xy 

-^/ 

a*-b* 
xl. 


(a'  +  2ab  +  b^){a^+b') 


12. ^-^ 


{x'  +  y*)(x*-2x'y'  +  7/) 

ax^-\-9ax-{-20a  '   6  +  5a;  +  ar^* 

,^     a*-14a«-51  ,^      a^-a^d' 

14.    — ■ — — -•  lb. 


a* -20" -15  (a-aby 

17.  At  two-thirds  of  a  cent  apiece,  what  will  b  apples 
cost? 

18.  James  is  x  times  as  old  as  George.     If  James  is 
a  years  old,  how  old  is  Georsje  ? 


34.   Illus.  1.  Change  to  an  equivalent  entire 


or  mixed  number. 


ac  —  bc  —  d  ,      d 

=  a  —  b 

c  c 


To  find  an  entire  or  mixed  number  equivalent  to 
a  given  fraxition,  divide  the  numerator  by  tJve  de- 
nom^inator. 


V 


96  A   FIRST  BOOK  IN  ALGEBRA. 


Illus.  2.  Change  to  equivalent  fractions  6  +  -,    b  — 


c  c 


a     be-}- a     ,       a—x      bc  —  a-{-x 

b-\--  = ,    b  — = 

G  c  c  c 


To  find  a  fraction  equivalent  to  a  given  mixed 
number,  multiply  the  entire  part  by  the  denojninator 
of  the  fraction,  add  the  numerator  if  the  sign  of  the 
fraction  he  plus,  subtract  it  if  the  sign  he  minus, 
and  write  the  result  over  the  denominator. 

How  may  an  entire  number  be  changed  to  a  fraction  hav- 
ing a  given  denominator  ? 

Illus.  3.  Change  -,    -,    and  —  to   equivalent   fractions 
b      d  ab 

having  a  common  denominator. 


a  X  ad 

a'd 

b  X  ad 

abd 

c  X  ab 

abc 

d  X  ab 

abd 

x_xd  _ 
ab  X  d 

dx 
abd 

To  change  fractions  to  equivalent  fractions  having 
a  common  denominator,  multiply  the  terms  of  each 
fraction  hy  such  a  number  as  will  mahe  its  denomi- 
nator equal  to  the  L,  C.  M.  of  the  given  denominators. 


FRACTIONS.  97 


Exercise  43. 

Change  tx)  equivalent  entire  or  mixed  numbers : 

5a;  —  ex  -t-  m  a^  -I-  .V^ 

X  '    x  —  y' 

mn  4-  gyi  +  x  a"^  —  b^ 

n  a-^b 

3.    ^-^  +  3.  11.    -M_. 

X  —  y  2a  — 1 

lib* 


a  +  6  3x4-1 

g     6a«6^c-9a^6'^H-3c  ^g  3x^  +  8x^  +  2 

3a26             '  '  x2  +  2x-l  * 

g     Gx^y'-h  J0a?'y'2-2m  ^^  2a3  +  3a'4- 10a  -  4 

2xy2               •  *            a2  +  3a  +  2 

^     x^  4-2x2-2x4-1  jg  2o^4-6a»-6a4-2 


x^  —  X— 1  a^4-a  —  1 

4-a2-2a4-l  ^g     3x^-5x«4-a;- 


Change  to  equivalent  fractions : 

17.  x  +  y-^^.  21.    ^^-x-2. 

^      x-y  3x-l 

18.  a-6-^.  22.    2a-l-^. 

a4-6  a4-3 

19.  -c  +  d-f^-^^     .        23.   a^_3a-h2--^^tA-. 

20.  ^Lh^  +  ar^_y2.  24.    2x^4- a;- 3  -  ^P^. 

x-y                 "^  ^                3x»4-l 


25. 
26. 

3a^  +  2         ^ 

a;2  _  ^  _|_  2 

^           2a-^  +  3 
a- -2a  4- 3 

27. 
28. 

a' 

a^  -  a  +  3 

a;^  +  a;  —  1 

98  A   FIRST  BOOK  IN  ALGEBRA. 

—  a-— a  — 1. 

-x'+x-l. 

Change    to    equivalent   fractions    having  a  common 
denominator : 

a'     xy     5ab\ 
''•    2'    T    "^  ^^• 

x^     am     3x^y 
^^'    3'    T'    ~5"  ^^' 

1^     5xy     x^ 
6y      2a      ay 

32.    2^,     ^,     ^.  36. 


5 

2 

3x         2ab 
x  —  y     x-  —  y- 

5  a         Sax 

3  + a' 
ft2 

a -3'     a' -9 
a'b 

a^  —  ah 

'    a?-h'' 

7m'     Sn      mn  """    lim  +  n^'    w?  —  rv 

37.    ^,     ? 

x"^  —  y^     x^  -{-  xy  —  2y'^ 

K  9. 

38. 


a  — 1  a  +  2  a  +  3 


40. 


41. 


a?-2a-lb     a^  +  a-Q'    a^-Ta  +  lO 

x-\-l  x  —  2  x-\-2 

x^-x-  6'    aj'-^  +  Ga^  +  S'    a;^  _^  a; -  12* 

a^  +  3a;4-2  a;^-a;  a^-6a;H-9 


42.    If  a;  quarts  of  milk  cost  m  cents,   what  will  one 
pint  cost  ? 


FRACTIONS.  99 

43.  Express  four   consecutive   numbers   of   which   a  is 
the  largest. 

44.  What  is  the  next  even  number  above  2m? 

45.  What  is  the  next  odd  number  below  4a  +  1  ? 

OPERATIONS    UPON    FRACTIONS. 
ADDITION  AND  SUBTRACTION. 

35.   Illus.  1.  Find  the  sum  of  ^'  ^^^,  and  -^^r^- 

9ar      9ar  9ar 


/    3a^-i-l\  ^ 6a'4-2a^-a-3a'-l 
"^V        9^2    y  9a^ 


6a^     2a^^-a 
9x*         9x2 

5a2_a-l 


9x2 


X  1  1 

Illus.  2.     Find  the  sum  of  -,  ,  and 

xy-f    x-y  y 

-y 


ic-y    V   2^/    y{^-y)    yi^-y)    y(x- 


xy-f     x-y     \    yj      y{x-y)     y{x-y)      y(x-y) 

^       2y      ^     2 
y(x  —  y)     x  —  y 

To  add  fractions,  find  equivalent  fractions  hav- 
ing a  common  denominator,  add  their  numerators, 
write  the  sum  over  the  common  denominator,  and 
reduce  to  lowest  terms. 


100  A   FIRST  BOOK  IN  ALGEBRA. 

Illus.  3.     Subtract  ^^^^^  from  ^Lzl^. 
a  —  2b  a 

a-2b     a-46^a^-4a6  +  45^       a'-4.ah  ^        46^ 
a  a  — 2b         a{a  —  2b)  a{a  —  2b)      a{a  —  2b) 

Make   a  statement   of  the   method   of   subtracting   one 
fraction  from  another. 


Exercise  44. 

Find  the  value  of : 

^     2a  ,  a     5a  ^     2x          ,  Sx 

5                  3  7i2             m^ 

_     2a;      3cc  ,  4a;  ,»       a;       a;  +  mn  ,  « 

3.    -^--r  +  — •  7. -^ +  2m. 

o        4        5  mn          3n 

.     2a  ,   Sx  „     264-a;  ,  56-4a; 

4. 8.    ' 

b       2m  3x              9x 

g     3m-a      b-2m  _  36-2a 
am  6m  ab 

10.      ^^^           ^^  13.    a;-4y      x^-4y- 

o?  —  b'^      a  4-  6  x  —  2y      x-  -{-2xy 

^^     2a  — Sb      2a  — b  x  —  7      x  +  4. 

a-2b        a-b'  '    a;  +  2      a;-3* 

12.    ^  +  4      a;  +  2  ^^^    (a;  +  2a)^           1 

a; +  5      a; -1-3  x^ — 8a^       a;  — 2  a 

^Q     a^  -^  x^f  -\-  y^  _^x^  -  x^f  -^  y^ 
x^-\-y^  Q^  —  y^ . 


FRACTIONS.  101 


17. 1 • 

?ttH-2     7n-f3     m  —  1 

18.     2(4a4-fe) 


15a«-156«     5(a-|-6)      3a -3Z) 

3ar'-3a^4-a;-l  ,  3a;»  + 3a:^-a;- 1 
•    3x»-3a^-aj  +  l      3ar»  +  3a:^  +  x+l 

..  1  1 


A\J. 

n«_7a-hl2      a*  — 5a +  6 

21. 

a_l          5-2a          a-2 
a_2      a^-ba  +  Q     a-3' 

22. 

.1          +         2a                      1          . 

a«  +  3a  +  2      a«  +  4a  +  3      a^  +  Sa-f  6 

23. 

a;_5     x  +  4       ar»_iF-20 

24. 

m  —  »i      p  —  m      n  —  p 
mn           mp           np 

25. 

a-\-b      a  —  c     c  —  b 
ab          ac           be 

26. 

x^  —  yz                    xr  —  xz                        z* 
{x  +  y){x-\-z)      {y^z)(y  +  x)      (z -\- x)  (z -\- y) 

27. 

?7i'  —  bx                   mx  —  b^                  mb  —  x' 

{m -\- b)  {m -\- x)      {b  +  x){b'\-m)      {x -\- m)  {x  ■{- b) 

28.  a;  is  how  many  times  y  ? 

29.  If  a  is  ^  of  a  number,  what  is  \  of  the  number  ? 

30.  If  a;  is  ^f  of  a  number,  what  is  the  number  ? 

3 1 .  Seven  years  ago  A  was  four  times  as  old  as  B.     If 
B  is  x  years  old,  what  is  A's  present  age  ? 


102  A   FIRST  BOOK  IN  ALGEBRA. 

36.   What  is  the  effect  of  multiplying  a  number  twice 
by  —  1  ^     xlow  many  signs  to  a  fraction  ? 

.J  2^cte  »3are'fuily  in  the  following  illustrations  the 
variety  of  "changes  that  may  be  made  in  the  signs 
without  changing  the  value  of  the  fraction: 


LLUS.     .     -_— _--^_-  — . 


In  case  the  terms  of  the  fraction  are  polynomials, 
notice  that  the  change  in  sign  affects  every  term  of  the 
numerator  or  denominator. 

T            o      a  —  b      b  —  a         b  —  a         a  —  b 
Illus.  2.     = = = 

^—y    y—^       ^—y       y—^ 

a  —  b-\-c_    b  —  c  —  a    _     b  —  c  —  a 

x-\-y-\-z      —x  —  y  —  z         x-^y-\-z 

a  —  b-\-c 


Illus.  3. 


—x—y —z 

When  the  denominator  of  the  fraction  is  expressed 
in  its  factors,  the  variety  of  changes  is  increased. 

Illus.  4. 

a  a  —a 


(x  —  y){m  —  n)      {y  —  x)  (n  —  m)  {y  —  x)  (m  —  n) 

—  a  —a 


{x  —  y){n  —  m)  {x  —  y){m  —  n) 

a__ a 

\y  —  x)  (m  —  n)  {x  —  y){n  —  m) 


FRACTIONS. 

Illus.  5. 

a  —  h                     b  —  a 

a-b 

{x^y){c--d)      {y-x){c-d) 

iy-x){d-c) 

a-b 

=  etc. 

103 


{y-x)(c—d) 

What  is  the  effect  on  the  value  of  a  fraction  of  changing 
the  signs  of  two  factors  of  either  denominator  or  numera- 
tor ?  Why  ?  If  the  signs  of  only  one  factor  of  the  denomi- 
nator are  changed,  what  must  be  done  to  keep  the  value  of 
the  fraction  the  same  ? 

Write  three  equivalent  fractions  for  each  of  the 
following  by  means  of  a  change  in  the  signs  : 

1.   ^.  3.   ^  +  y  5    x-y-z 

mn  a  —  b  c-f-d  —  a 

2    ^^^-  4      1  —  a?  /.      3  g  —  a;  -f  y' 

2f'  '    3c -a**  '    m^-\-2c'd-z 

Write  six  equivalent  fractions  for  each  of  the  fol- 
lowing : 

J    X g  a  —  m 


(a  — 6)(m  — ?i)  {c  —  d){x  —  y) 

8  Sa^bc  1^  2c-\-d 


(2x-y){a-z)  (^a  -  b -{- c) (x - z) 

Write  as  many  equivalent  fractions  as  possible  for 
each  of  the  following  : 

11.   ^ 13.  "'^ 


{a  —  b)  (x—y)  {m—n)  (2m  — n)  (c—x)  {d—y^) 

12.    9l 14.  (x-y){2a-b) 

{c—d){m  —  x){y—b)  (m  —  x)ia  —  c)(d  —  y) 


104  A    FIRST  BOOK  IN  ALGEBRA. 

Illus.  1.     Find  the  value  of 


x-fl      1  —  X      x^  —  1 
1  1  X  1.1  X 


X  -{-1      1  —  X      a^  —  1      cc-j-l      X  —  1      x^  —  1 

X  —  1  X  -\-l  X 


x^-1      x^-l      x'-l      x^-l 


Illus.  2.    Find  the  value  of --\- 


{a-b){b-c)     {b-a)(a-c) 

+ L__. 

{c  —  a){c  —  b) 

1       +,.     A     ,+       ^' 


(a  —  b)(b  —  c)       (6  —  a)  (a  —  c)       (c  —  a)  (c  —  b) 

1  1       +       1 


(a  — 6)(&  — c)       (a  — 6)(a— c)       (a—c)(b  —  c) 
a  —  c  b  —  c 


(a  -  6)  (6  -  c)  (a  -  c)       (a  -  6)  (6  -  c)  (a  -  c) 

a  —  6 


(a  —  b){b  —  c)  (a  —  c) 
2a-2b  2 


(a  — 6)  (6  —  c)  (a  — c)       (6  — c)(a  — c) 
Exercise  46. 


Find  the  value  of : 


1.  4^+^+  1 


2. 


ar  — 4      2  — a;      2  +  a; 
3a      .       1  1 


a2_93-a      3  +  a 
3.    m 


2        am^  am-         2aV 


a  —  m      m-\-a     m^  —  a^ 


FRACTIONS.  105 


20a- 4  ,       3  6 

4.    -r-^ r  + 


4a«_l      l_2a      l  +  2a 
'   x  —  y      f  —  o^      a?-\-xy-\-y^ 

6(a=^  _  1)  ^  2(1  -  a)      3(1  +  a) 

1 1 116 -7y^ 

*   5(36-/)      2(36  +  2/2)      10(y*-96») 

3.1 


8. 


(l_.a;)(3-a;)      (2-a;)(a;-3)      (a;-l)(a;-2) 


9.   ^ +     1  «-^ 


10. 


(a  — 6)(6  — c)      c  —  a      {c  —  a){c  —  h) 

1  .1  1 


{z-a){z-y)      {a'-z){a-y)      (y-z)(y-a) 

11.    i ^ ^ t 

{l-x){l-y)       (x-l){y-x)       {y-l){x-y) 

12.  a-     «' 


a+1      1— a 
13.    1  — a  +  a^- a^  + 


2         «3    •  ^ 


14. 


1  +  a 
1  2 


aj«-7a;  +  12     a:*-6x  +  8     aj*-5a;  +  6 

15.  What  will  a  pounds  of  rice  cost  if  6  pounds  cost  43 
cents? 

16.  a;  is  J  of  what  number  ? 

5 

17.  w  is  -  of  what  number  ? 


18.    V  is  -  of  what  number  ? 
^      6 


106  A    FIRST  BOOK  IN  ALGEBRA. 


MULTIPLICATION    AND    DIVISION. 

37.   iLLus.         ^xc  =  ^,        i^^xa5  =  i^3 
y  y  oa^b&  oacr 

A  fraction  is  multiplied  by  multiplying  the  numer- 
ator or  dividing  the  denominator,* 


Exercise  46. 

Multiply : 

.1.    ^^  by   X.  6.    ^  by   3a:. 

b^  2  mn 

2-    77^-i  by   f-.  7.   -^ by   a;. 

1  a^ 

4.   ^-5^'  by   3^2,.  9.   §^±^  by  :«^-/. 

9  ari/  ^  —  y 

3  mw   1       ^2  ,_a^  —  4a  —  21,       ^      ^ 

2  »y  a-  —  3  a  —  10 

11.  x  +  2/ 7^  by   ^  +  2/- 

a^  +  2/ 

12.  a- 5  +  ^^  by   a -6. 

a  — 6 


1 2^ 

a' -2/ +  2;      {x-y)--z^ 


13. — T-  +  T7 TT^ 2  by   ar  -  a;?/ -  a;2;. 


14.    ^ —^ by   ac^-bc^c\ 

a-b  +  c      {a-^cf-b^     ^ 

*  In  arithmetic,  which  of  these  two  methods  did  you  find  would  apply- 
to  all  examples  ?  When  either  method  may  be  applied  to  any  given  exam- 
ple, which  is  preferable,  and  why  ? 


FRACTIONS.  107 

15.  How  many  units  of  the  value  of  \  are  there  in  2  ? 

16.  How  many  units  of  the  value  of  -  are  there  in  5? 

17.  How  many  sixths  are  there  in  4a ? 

oo    I  ^V^       J      ^      5ahc      o  5ahc 

^   fraction   is   dividexl  by  dividing   the  numerator 
or  multiplying  the  denominator  * 


Exercise  47. 

Divide : 

1.  ^  by   ax.  5.    -^   by   3a;. 

2.  ?^  by   2x.  6.   ?4=#^   by   3a:. 

2xy      ^  2mH         ^ 

3.  l^cd^  by   6cd3.  7.    ^tT''^  by   2a6. 

.     6a;*y   ,       ^  „     5(a2-62)    ,              . 

4.  — -   by   9mw.  8.    --^^ by   a  —  6. 

9.        ^  +  ^^  by  ^^-n\ 


7>l' 


2  mn 


10.  ^±|^^by   x-2. 

11.  ax-f  &^'  +  a  +  ^  +  ^^-^  by   a  +  &. 

3a; 

12.  Multiply  ^~f^"^^  by  a; -  1. 

ar  4-  3aj  —  4 

*  See  foot-note  on  opposite  page. 


108  A   FIRST  BOOK  IN  ALGEBRA 


6ax^ 


13.    Divide  ~— ^  by   10 abc. 


14.  Multiply  1  +  ^   by   2ac. 

4:X^y 

15.  Divide  ^^'^~^^'  by   Ax-\-Ay. 

16.  Divide  ^^   by   Qaxy. 

3  mir 

17.  A  can  do  a  piece  of  work  in  2\  days,  and  B  in  2\ 
days.  How  much  of  the  work  can  they  both  together  do 
in  one  day? 

18.  If  C  can  do  a  piece  of  work  in  m  days,  and  D  works 
half  as  fast  as  C,  how  much  of  the  work  can  D  do  in  one 
day? 


f>fx    T  ^      X      a      ax 

39.   Illus.  1.     -  X  -  =  — 

2/4     4?/ 


Illus    2      ^^^V^^Oah' _2xyz 


J  o      ab~ax  —  2b'\-2x      b^x  -f  bx^  -\-a^ 


W-x^  a  -  2 

(a  -  2)  (6  -  x)x{b^+bx  +  x") 


—  X 


■  {b  -  x)  {b'  -\-bx-^  x^)  {a  -  2) 

To  multiply  one  fraction  by  another,  multiply  the 
numerators  together  for  the  numerator  of  the  prod- 
uct and  the  denominators  for  its  denominator,  and 
reduce  to  lowest  terms. 


FRACTIONS.  109 


Exercise  48. 


Simplify : 

18  ar^/z     2Sab(^' 

9a^b^       6a?f      24  mVa; 
*   %n3?y^^2amhi        90a6«  * 

3     a^-16y^^     2y  ^     ^^^x-6  ^  x^+3x-4 

xy  —  4:y^      x-{-4:y  '   x^-^x  —  2     x^  —  2x  —  'S 

'       26     ^    9a2-62   '  •     a2+a-2  ^      a  +  3      ' 

g     a6d-f  c(?^     acd-6c^  ^     a.-* -64  a;      9a;'-4 

a*d  — a6c     a6*4-6cd  *   33^^  —  20;     a:^  — 4a; 

2aa;  —  4  V      wny  +  ay^  m^  —  25         m^  —  27  m 

mn^-{-any     a-x  —  2ay  m^-\-2m—W     m^  —  bm 

a»  —  a^y^afax  +  xy  —  ^a  —  3y 
a;_3         ^  a»-|-?/8 

^2     ar'-|-ar^-a;-l  ^     ar^  +  2a;-3 


a;2-f6a;  +  9        2a;»  +  4ar' +  2a 

^^    a»-a«-,a-t-l^     a^-f3a4-2 
a«  +  4a4-4        3a^-6a2  +  3a 

14.      wiH (n . 

\         in  —  nj\        m  +  n/ 

15.  A  grocer  purchased  y  pounds  of  tea  for  $25. 
Another  grocer  purchased  4  pounds  less  for  the  same 
money.     What  was  the  price  per  pound  which  each  paid  ? 


110  A   FIRST  BOOK  IN  ALGEBRA. 

16.  What  is  ^  of  ?? 

3        c 

17.  Two  numbers  differ  by  28,  and  one  is  eight-ninths 
of  the  other.     What  are  the  numbers  ? 

40.   iLLus.  ^.^-J-^y^"! 

c      y      c      X      ex 

To  divide  one  fraction  by  another,  invert  the  divi- 
sor and  proceed  as  in  multiplication. 


Exercise  49. 

Simplify : 

,     IS  a^b-c       Sa*b  ^     2am^n^     Sa^mn^ 

1. :r^r--i-z — 4. 


S5oi^yh      5x^yz  Sx'yz^        Ixy^z^ 

8m^n      3a6^m  oi?  v?-\-hx 

g     4a;VV  .  3a;Vg^  ^     x-^  .  x^ -?>x 

3a'b^c'  '  4aW'  *    x-S  '  x'-5x 

x-2  ^    a;'^-9a;-f20       a;^-4a;  +  3 
•    a;_7  •  x2_i()^^21  *  3^-5x-{-4.' 


8. 


a2_7q^   ^  a^4-l()a  +  24  .  a^-7a  +  6 


^       a,4-5     .a^  +  ^%^  «' -  6* 


(a -6)2     a2-62      (a +  5)3 

10.  ^  —  y\^^-y*  :  (^  +  y)^ 

*    a^  +  /      (a;_y)3  •      a;_2, 


11. 


2a-S       a^-\-5a-\-6       a^-5a  +  4: 


a24.2a-15      a^-4a-\-3     a^-^Sa  +  15 


,-     iB2_9£c  +  20  .  a^-7a;4-12^ 


a^_5a;_14       a;2-a;-42       a.-2  +  llaj  +  30 


FRACTIONS.  Ill 

20* 


a'H-2a— 8       ah-\-b    '  a^  —  a  —  L^ 

m  —  2        y>i^  —  5  m  —  14        4  m''     ^     2  ??i7i  —  14  n 
2  m  — lA:        2  m*  +  4  m         m^  —  4  *  m^  —  5  m  —  14 

-(i^--)(iTT-')-e^-'X'-:Tfi)- 

■•^r^.)('-ri;)-(•-Bl)(■-ffl> 

17.  How  many  times  7  is  21  ? 

18.  How  many  times  -jS^  is  f  ? 

19.  How  many  times  -  is  -  ? 

d     4 

20.  How  many  times  -^  is  ^^^,  ? 

7  mar      4  a?M7i^ 


? 


What  may  be  done  to  a  fraction  without  changing  its 
value  ?  How  multiply  a  fraction  ?  When  is  a  fraction  in 
its  lowest  terras  ?  How  reduce  a  fraction  to  lowest  terms  ? 
How  divide  one  fraction  by  another  ?  How  add  fractions  ? 
How  divide  a  fraction  ?  What  is  the  effect  of  increasing 
the  denominator  of  a  fraction?  What  is  the  effect  of 
dividing  the  numerator  of  a  fraction  ?  What  is  the  effect 
of  subtracting  from  the  denominator  of  a  fraction  ?  How 
multiply  two  or  more  fractions  together  ?  What  is  the 
effect  of  adding  to  the  numerator  of  a  fraction  ?  What  is 
the  effect  of  multiplying  a  fraction  by  its  denominator? 
What  changes  in  the  signs  of  a  fraction  can  be  made  with- 
out changing  the  value  of  the  fraction  ?  How  change  an 
entire  number  to  a  fraction  with  a  given  denominator  ? 


112  A   FIRST  BOOK  IN  ALGEBRA. 

INVOLUTION,  EVOLUTION,  AND  FACTORING. 

41.   iLLus.  1.     y=-x^  =  -. 

What  is  the  square  of  —  ?     What  is  the  cube  of  -?   of 

_  ?^  ?     What  is  the  square  root  of  ^  ?     What  is  the 
xy  9y^ 

cube  root  of ?     How  find  any  power  of  a  fraction  ? 

276y  .  ^ 

How  find  any  root  of  a  fraction  T 

^  o      /wi  ,  1\  /^m      1\      m^      1 

Illus.  2.       -  +  - =~~""2' 

\n      xj\n      xj       n-      xr 

Illus.  3.     /^^-3V--6^  = -'-  — +  18. 

\y     J\y     J    y'     y 

Illus.  4.     Factor ^• 

cr     x'^ 

!^^t=(^j^f\(^j^y\fk-.y\ 

a*      x^      \o?      x^)\a     xj\a     xj 


Illus.  5.     Factor  x"^  -\-x^  -^\ 


x*j^x'  +  \  =  {x'-^^Y. 


Exercise  60. 

Find  by  inspection  the  values  of  each  of  the  following : 

■    \ab\l  '    \  3a'b    J 

fa'by  (  Bx'yia  +  hyY 

'    [4.y)'  '   \  2ah\^-yyj' 

3     f      2^y  Y  6     (  3«m^(2a  +  36)^Y 

I.     3a2mV  '    I  4xV(m'-'n)V 


FRACTIONS.  113 

./m^i.  18.  f?-£y"+5Y 

20     ^^2a:_3j(^^^2/Y 
yzV3        2a;       iti-z) 

«■  (^^)(^S)(!-i){|-i> 
22  ^^+l2lV?  +  ^V?-^Y 


114  A   FIRST  BOOK  IN  ALGEBRA. 


Factor : 

29. 

x' 

a' 

30. 

a' 

of 

31. 

a' 

b* 
x^' 

32. 

33. 

x' 

?f 

27  a 

?      c' 

34. 

81a*       xY 

b'        625z' 

35. 

vi'^2m      ^^ 

y-      y 

36. 

^  _2x_^ 
or       a 

37. 

f  +  2+i, 
4              2/' 

38. 

a?^x^ 

39.  A  dog  can  take  2  leaps  of  a  feet  each  in  a  second. 
How  many  feet  can  he  go  in  9  seconds  ? 

40.  How  many  weeks  would  it  take  to  build  a  stone 

wall  if  -  of  it  can  be  built  in  one  day  ? 
a 


COMPLEX  FRACTIONS. 


42. 

Illus. 

'-\ 

a 

'-\ 

a 

X 

d     ' 

y 

y 

b 
c 

A  complex  fraction  is  one  which  has  a  fraction  in 
one  or  both  of  its  terms. 

What  is  a  simple  fraction  ?     What  is  the  effect  of  multi- 
plying a  simple  fraction  by  its  denominator  or  by  any  mul- 


FRACTIONS.  115 

tiple  of  its  denominator  ?     By  what  must  the  numerator  of 

the  complex  fraction   be   multiplied   to  make  the 

-  +  — 
d      ah 

numerator  an  entire  number?  By  what  multiply  the 
denominator  to  make  it  an  entire  number  ?  If  both  numer- 
ator and  denominator  are  multiplied,  under  what  conditions 
will  the  value  of  the  fraction  remain  the  same  ? 


Illus.  1.     Simplify 

c       x_ 

d      ab 


ax 
d     ab 


Illus.  2.     Simplify  — --^ i — 


1—x     1+x 
1 


X(l-x)(l-f«) 


1—x     l-\-x  _1  4-a;  — l  +  aj_2a;_ 

1—x     1+3; 

To  slTYiplify  a  complex  fraction,  multiply  each 
term  of  the  complex  fraction  by  the  L.  C.  M.  of  the 
denominators  of  the  fractions  in  the  terms,  and 
reduce. 


116  A    FIRST  BOOK  IN  ALGEBRA. 

Exercise  61. 


Simplify : 


1. 

^+f 

1-' 

2. 

y 

y 

3. 

X 

c 

4. 

-h 

1-1 

X 

a      b 
b     a 

^'  7. — r  15.  1 


l+i 

X 

z 

y 

X 

a 

b 

y 

X 

a6_ 

-?>d 

c 

3c- 

ab 

10. 


a-fl 
1  +  ct  -f  g^ 

07-3  ^ 


n.  i^ 


ic  — 4 


a  a 


a4-6      a  — & 

12. 

2a 


a^-b^ 
a-\-l      a  —  1 
13     ^-^      ^  +  ^ 

a— 1      a+l 

6.  1±^.  oM:!^ 

14.      ^'-'' 


d^-abJr  b^ 
a  —  b 

1 


'-.4. 


8.    -.  16. 


a-\-2b      a 
a4-b       b 


FRACTIONS.  117 

17.  How  many  pounds  of  pepper  can  be  bought  for  y 
dollars,  if  x  pounds  cost  2x  dimes  ? 

18.  How  many  inches  in  y  feet?     How  many  yards? 

19.  A  man  bouglit  x  pounds  of  beef  at  x  cents  a  pound, 
and  handed  the  butcher  a  ^/-dollar  bill.  How  many  cents 
change  should  he  receive  ? 

20.  X  times  I  is  how  many  times  m  ? 

Exercise  52.     (Review.) 

1.  Divide   o^- 19a;  +  Ga:' +  20 -f  8*- +  16a;^- ojP  -  lire* 
by  ar^  +  4-3a;  +  2ar». 

2.  Find   two   numbers   whose   sum  is  100  and  whose 
difference  is  10. 

3.  Findthe  G.C.F.  of  a;*-l,    x" +  x* -x^ -x"  +  x-\-l, 
and  ar  —  X  —  2. 

4.  Factor  a:'«-14a;»y  + 49/,     a»-6^     3a2-3a-216 

5.  Reduce  _(2a  +  26)(o'-6'^)      ^^  ^^^^^^  ^^^.^^ 

(a2H-2a6  +  6'^)(a-6) 

6.  Factor  15^^-/-. 

a*6*      81  / 

7.  If  a?  and  y  stand  for  the  digits  of  a  number  of  two 
places,  what  will  represent  the  number? 

8.  Find  the  L.  CM.  of   a--5a-\-  6,    a^  -  16,     a-  -  9, 
a* -7a +  12,   and   ar-^. 

9.  If  one  picture  costs  a  cents,  how  many  can  be  bought 
for  X  dollars  ? 


118  A    FIRST  BOOK  IN  ALGEBRA. 

Simplify : 

10     ^-3       2(1-^)        x-1 

11.       1 2 ^  1 


(2 -a;)  (3 -a;)       (a;-l)(a;-3)       {x-l){x-2) 

a^ -\-xy  ^  x^ —  Sx^y -\-Sxy^ —  y^      2xy  —  2y^ 

14,    ^x 7, ^ '• '—iz 

x  —  y  or  —  2/  o 


13.    [l+hia+b)-(^-±' 


1  .  X 


•)~m 


X 


1*- '  1+^j  1+1 

X  — 


?  +  &__!      1  +  6      1+6! 

^^     0      a  a  a"* 

15.    — X 


h-     h  ha 

16.  Prove  that 

3  (a  -h  6)  (a  +  c)  (6  +  c)  =  (a  +  6  +  c)^  -  (a^  +  6'  +  c')  • 

17.  How  many  sevenths  in  6  a;?/ ? 

18.  Divide  2x^ -  — -\-2\^ -2x'^ -x  by  3a.'2  +  a;. 

1^ 


EQUATIONS. 


43.  Illus.  1.     27  +  lOx  =  13  a;  -f-  23. 

Illus.  2.  _h-^  =  _^  +  ^. 

a;      2       a;       4 

An  equation  is  an  expression  of  the  equality  of  two 
numbers.  The  parts  of  the  expression  separated  by  the 
sign  of  equality  are  called  the  members  of  the  equation. 

The  last  letters  of  the  alphabet  are  used  to  represent 
unknown  numbers,  and  known  numbers  are  represented  by 
figures  or  by  the  first  letters  of  the  alphabet. 

44.  Illus.  1.    5a;+20=105.        Illus.  2.    3a:-18  =  42. 

5a;  =  85.  3a;  =  60. 

Illus.  3.  ^=^*  Illus.  4.  7  a;  =49. 

a;  =  24.  a;  =  7. 

Any  changes  may  be  made  in  an  equation  which 
do  not  destroy  the  equality  of  the  members. 

Name  some  of  the  ways  in  which  such  changes  may 
be  made. 

45.  Illus.  1.  a;  4-  6  =  a.  Subtracting  h  from  each  mem- 
ber,  x  =  a  —  b. 

119 


120  A    FIRST  BOOK   TN  ALGEBRA. 

Illus.  2.    x  —  b  =  c.    Adding  b  to  each  member,  x  =  c-{-h. 

In  each  of  these  illustrations  b  has  been  transposed 
(changed  over)  from  the  first  to  the  second  member, 
and  in  each  case  its  sign  has  been  changed.     Hence, 

Any  term  'may  be  transposed  from  one  tnember  of 
an  equation  to  the  other  provided  its  sign  be  changed, 

46.  Illus.     ctb  + x  _b^-x^x-b_ab -x^ 

b'  a'b  a'  W 

Multiply  both  members  by  arW, 

a^b  +  a-x  —  b^-{-bx=  b'x  —  b^  —  a^b  -f  a^x. 

To  clear  an  equation  of  fractions,  multiply  each 
memher  of  the  equation  by  the  L.  C.  M.  of  the  denom- 
inator's of  the  fractional  terms. 

47.  Illus.  1.    Solve  a; -f  4 -|- 2(ic-l)=  3a;+4-(5a;-8). 

a;  +  4  +  2a;-2  =  3a;  +  4-5aj  +  8. 
Transposing,  a;  +  2a;  — 3a;  +  5a;  =  4  +  8  — 4  +  2. 

5a;  =  10. 

x  =  2. 

Illus.  2.     Solve  ^-^+^-^^li^  +  ^  =  ?l-^^^. 
8  3  2      2  6 

Multiply  by  24, 

3(5rc  +  3)  -  8  (3  -  4a;)  +  12a;  =  372  -  4  (9  -5a;). 
15a;  +  9  -  24  +  32a;  +  12a;  =  372  -  36  +  20a;. 
Transposing,  15a;  +  32a;  -f  12a;  -  20a;  =  372-36-9  +  24. 

39  a;  =  351. 
a;=9. 


EQUATIONS.  121 

To  solve  an  equation,  clear  of  fraxitwns  if  neces- 
sary, transpose  tJie  terms  containing  the  unknown 
number  to  one  member  and  the  known  terms  to  the 
other,  unite  the  terms,  and  divide  both  members  by 
tlie  coefficient  of  the  unknown  number. 

Exercise  53. 

Solve : 

1.  22-6x=34-12a?.  4.   5a;-21  =  7a;-f  5. 

2.  5a;-4=10a;H-ll.  5.    18a:- 43  =  17  -  6a;. 

3.  23-8a;  =  80-llfl?.  6.    18  -  8a;  =  12a;- 87. 

7.  9a; -(2a; -5)  =  4a; +  (13  + a;). 

8.  lox- 2(5a;-4)-39  =  0. 

9.  12a; -18a; +17  =  8a; +  3. 

10.  21a;-57  =  6a;-14a;  +  30. 

11.  5a;-27-lla;+16  =  98-40a;-41. 

12.  14(a;-2)  +  3(a;  +  l)  =  2(a;-5). 

13.  6(23-x)-3a;  =  3(4a;-27). 

14.  3(a;-l)-2(a;-3)  +  (a;-2)-5  =  0. 

15.  (a;  +  5)(a;-3)  =  (a;  +  2)(a;-5). 

16.  (x  +  4)(a;  +  7)  =  (x  +  2)(a;  +  ll). 

17.  (x-l)(a;  +  4)(x-2)  =  a;(a;-2)(a;  +  2). 

18.  (a;-5)(x  +  3)-(x-7)(a;-2)-2(x-l)  =  -12. 

19.  (x  +  2)*-(x-l)«  =  5(2x  +  3). 

20.  (2x  +  3)(a;  +  3)-14  =  (2x  +  l)(x  +  l). 

21.  («  +  l)H(x-6)»  =  2(«  +  5)'^. 


122  A    FIRST  BOOK  IN  ALGEBRA. 

22.  7x  —  15-{-4:X  —  6  =  4:X—9  —  9x. 

23.  Divide  the  number  105  into  three  parts,  such  that 
the  second  shall  be  5  more  than  the  first,  and  the  third 
three  times  the  second. 

24.  A  man  had  a  certain  amount  of  money ;  he  earned 
four  times  as  much  the  next  week,  and  found  $  30.  If  he 
then  had  seven  times  as  much  as  at  first,  how  much  had 
he  at  first  ? 

25.  How  many  fourths  are  there  in  7fl?  ? 

26.  How  long  will  it  take  a  man  to  build  x  yards  of  wall 
if  he  builds  z  feet  a  day  ? 

27.  6a;-^4^  =  |  +  28. 

o  o 

9 


28     ^a^-l      ^  +  12      ,^_2x 
^^'   "~5~      ~3~     ^^-3 


29.    a;-^  +  25  =  |  +  ^  +  21. 
3     ^21  7 


__     8  — 5a;  ,  5a;  — 6      7a;  +  5_^ 

32     3  +  50^      ^^  +  2_^, 
32.  4+2      ~^' 


33. 


34. 


2a;-l      13      5a; 


3  42  6 

l-llo;     7a;^  2       8a;-15 
7  13      13  3      ' 


EQUATIONS.  128 


35   2__A  =  1_A 

'   X     2x     24.     Zx 

36.   l  +  ±=Ll  +  ±. 
4     3a;     36     6a; 

38.    ?^--?  +  4  =  A  +  -^-h2H- 
5a;         X  2x     4a; 


39. 


40. 


x  +  9     2  — a;^a;-h5 
11  5     ~     7    ' 

a;  4-4      4  —  a;     x-\-l 


5  7  3 

41.  |(x-6)-^(a;-l)  =  /^(4-a;)-A- 

42.  i±^*_l^_K8-^)  =  ^^  +  7. 

43.  How  long  will  it  take  a  man  to  walk  x  miles  if  he 
walks  15  miles  in  6  hours  ? 

44.  What  is  the  interest  on  m  dollars  for  one  year  at  5 
per  cent  ? 

45.  What  are  the  two  numbers  whose  sum  is  67,  and 
whose  difference  is  25  ? 

46.  What  is  the  interest  on  b  dollars  for  y  years  at  4  per 
cent? 

47.  (c-f  a)a;H-(c-6)a;  =  c*. 

48.  (a  — 6) a;  4- (a -1-6) a;  =  a. 

49.  2a;-|-a(a;-2)  =  aH-6. 

50.  b{2x-a)-a^  =  2x(a  +  b)-3ab. 


124  A    FIRST  BOOK   IN  ALGEBRA 


1.    a^-{-c-  = 1 

a       c 

52.  b'-x^-^2bx  =  {b'-{-x){b'-x). 

53.  x^-{-4.a^  +  a'  =  (x  +  ay. 

54.  b\x  -  6)  +  ci^{x  -a)  =  abx. 

__     2a;  +  5      2x-4:  ^„        2  +  9a;  2a;  +  9 

Oo.     = &7»     =  • 

5aj-l-3     5x-6  3(6a;4-7)      35  +  4a; 

56.    6^±5^3^zii.  58.    -^ ^  =  0. 

2a;-3       ic  +  1  2-5x     l-3a; 

gg     2(30^4-4)      ^^2(19  +  0^) 

l_|_2a;  a;  +  12 

60.    1  ^'  ^ 


ic4-2      aj  +  6 


1  —x^      x  —  1      x-{-l 

62.    2  +  3^^5^20^-4. 
l-x  x-\-2 

63.  _A_  +  _^=_^i_  +  _A_. 

6-\-2x     x-hl      2  +  2a;     3  +  a; 

64     2(1-20^)      1^1-30^ 
l-3a;         6      l-2a; 


65. 


66. 


a;  — 8_     X     _x  —  9     x-\- 1 
a;  — 6     aj  — 2     a;— 7      a;  — 1 

a;-f-5     x-\-6      x-\-2      x-\-S 


x-i-S     x-\-9     x-\-5     x  +  6 

67.  Find  three  consecutive  numbers  whose  sum  is  81. 

68.  A's  age  is  double  B's,  B's  is  three  times  C's,  and  C 
is  y  years  old.     What  is  A's  age  ? 


EQUATIONS.  126 

69.  How  many  men  will  be  required  to  do  in  a  hours 
what  X  men  do  in  6  hours  ? 

70.  Find  the  sum  of  three  consecutive  odd  numbers  of 
which  the  middle  one  is  4a; -fl. 

Exercise  54. 

1.  In  a  school  of  836  pupils  there  is  one  boy  to  every 
three  girls.     How  many  are  there  of  each  ? 

2.  Divide  253  into  three  parts,  so  that  the  first  part  shall 
be  four  times  the  second,  and  the  second  twice  the  third. 

3.  The  sum  of  the  ages  of  two  brothers  is  44  years,  and 
one  of  them  is  12  years  older  than  the  other.  Find  their 
ages. 

4.  Find  two  numbers  whose  sum  is  158,  and  whose  dif- 
ference is  86. 

5.  Henry  and  Susan  picked  16  quarts  of  berries. 
Henry  picked  4  quarts  less  than  three  times  as  many  as 
Susan.     How  many  quarts  did  each  pick  ? 

6.  Divide  127  into  three  parts,  such  that  the  second 
shall  be  6  more  than  the  first,  and  the  third  four  times 
the  second. 

7.  Twice  a  certain  number  added  to  four  times  the 
double  of  that  number  is  90.      What  is  the  number  ? 

8.  I  bought  some  five-cent  stamps,  and  twice  as  many 
two-cent  stamps,  paying  for  the  whole  81  cents.  How 
many  stamps  of  each  kind  did  I  buy  ? 


126  A   FIRST  BOOK  IN  ALGEBRA. 

9.  Three  barns  contain  58  tons  of  hay.  In  the  first 
barn  there  are  3  tons  more  than  in  the  second,  and  7  less 
than  in  the  third.     How  many  tons  in  each  barn  ? 

♦  10.  If  I  add  18  to  a  certain  number,  five  times  this 
second  number  will  equal  eleven  times  the  original  number. 
What  is  the  original  number  ? 

11.  In  a  mixture  of  48  pounds  of  coffee  there  is  one- 
third  as  much  Mocha  as  Java.  How  much  is  there  of 
each? 

12.  The  half  and  fifth  of  a  number  are  together  equal  to 
56.     What  is  the  number  ? 

13.  What  number  increased  by  one-third  and  one-fourth 
of  itself,  and  7  more,  equals  45  ? 

14.  What  number  is  doubled  by  adding  to  it  three- 
eighths  of  itself,  one-third  of  itself,  and  14  ? 

15.  A  grocer  sold  27  pounds  of  sugar,  tea,  and  meal. 
Of  meal  he  sold  3  pounds  more  than  of  tea,  and  of  sugar 
6  pounds  more  than  of  meal.  How  many  pounds  of  each 
did  he  sell  ? 

16.  A  son  is  two-sevenths  as  old  as  his  father.  If  the 
sum  of  their  ages  is  45  years,  how  old  is  each  ? 

17.  Two  men  invest  $2990  in  business,  one  putting  in 
four-ninths  as  much  as  the  other.  How  much  does  each 
invest  ? 

18.  In  an  election  47,519  votes  were  cast  for  three  candi- 
dates.     One  candidate  received  2061  votes  less,  and  the 


EQUATIONS.  127 

other  1546  votes  less,  than  the  winning  candidate.     How 
many  votes  did  each  receive  ? 

19.  John  had  twice  as  many  stamps  as  Ralph,  but  after 
he  had  bought  65,  and  Kalph  had  lost  16,  they  found  that 
they  had  together  688.     How  many  had  each  at  first  ? 

20.  Find  three  consecutive  numbers  whose  sum  is  192. 

21.  If  17  be  added  to  the  sum  of  two  numbers  whose 
difference  is  12,  the  result  will  be  61.  What  are  the 
numbers  ? 

22.  Divide  120  into  two  parts  such  that  five  times  one 
part  may  be  equal  to  three  times  the  other. 

23.  Mr.  Johnson  is  twice  as  old  as  his  son ;  12  years 
ago  he  was  three  times  as  old.     What  is  the  age  of  each  ^ 

24.  Henry  is  six  times  as  old  as  his  sister,  but  in  3  years 
from  now  he  will  be  only  three  times  as  old.  How  old  is 
each  ? 

25.  Samuel  is  16  years  older  than  James ;  4  years  ago 
he  was  three  times  as  old.     How  old  is  each  ? 

26.  Martha  is  5  years  old  and  her  father  is  30.  In 
how  many  years  will  her  father  be  twice  as  old  as  Martha  ? 

27.  George  is  three  times  as  old  as  Amelia;  in  6  years 
his  age  will  be  twice  hers.     What  is  the  age  of  each  ? 

28.  Esther  is  three-fourths  as  old  as  Edward ;  20  years 
ago  she  was  half  as  old.     W^hat  is  the  age  of  each  ? 

29.  Mary  is  4  years  old  and  Flora  is  9.  In  how  many 
years  will  Mary  be  two-thirds  as  old  as  Flora  ? 


128  A   FIRST  BOOK  IN  ALGEBRA. 

30.  Harry  is  9  years  older  than  his  little  brother ;  in 
6  years  he  will  be  twice  as  old.     How  old  is  each  ? 

31.  Divide  2 a;*  4- 27a;/  — 81 2/^   by   x-\-3y. 

32.  Prove  (a'-\-ab  +  by -{a'-ab-\-by=4:ab{a^-\-b~). 

33.  Find  the  value  of  ^^^^  +  -^^  - ^,~^^% 

y         x-y      x'y-f 

34.  Solve  ^^^-3a.=?^±-^-14. 

2  3 

35.  Mr.  Ames  has  $132,  and  Mr.  Jones  $43.  How 
much  must  Mr.  A.  give  to  Mr.  J.  so  that  Mr.  J.  may  have 
three-fourths  as  much  as  Mr.  A.  ? 

36.  A  has  $101,  and  B  has  $35;  each  loses  a  certain 
sum,  and  then  A  has  four  times  as  much  as  B.  What  was 
the  sum  lost  by  each  ? 

37.  A  certain  sum  of  money  was  divided  among  A,  B, 
and  C ;  A  and  B  received  $  75,  A  and  C  $  108,  and  B  and  C 
$  89.     How  much  did  each  receive  ? 

Suggestion.     Let  x  equal  what  A  received. 

38.  Mary  and  Jane  have  the  same  amount  of  money. 
If  Mary  should  give  Jane  40  cents,  she  would  have  one- 
third  as  much  as  Jane.  What  amount  of  money  has 
each  ? 

39.  An  ulster  and  a  suit  of  clothes  cost  $43;  the  ulster 
and  a  hat  cost  $  27 ;  the  suit  of  clothes  and  the  hat  cost 
$  34.     How  much  did  each  cost  ? 


EQUATIONS.  129 

40.  John,  Henry,  and  Arthur  picked  berries,  and  sold 
them ;  John  and  Henry  received  $  4.22,  John  and  Arthur 
$3.05,  Henry  and  Arthur  $3.67.  How  much  did  each 
receive  for  his  berries  ? 

41.  A  can  do  a  piece  of  work  in  4  days,  and  B  can  do  it 
in  6  days.     In  what  time  can  they  do  it  working  together  ? 

Suggestion.  Let  z  equal  the  required  time.  Then  find  what  part 
of  the  work  each  can  do  in  one  day. 

42.  Mr.  Brown  can  build  a  stone  wall  in  10  days,  and  Mr. 
Mansfield  in  12  days.  How  long  would  it  take  them  to  do 
it  working  together  ? 

43.  Mr.  Richards  and  his  son  can  hoe  a  field  of  corn  in 
9  hours,  but  it  takes  Mr.  Richards  alone  15  hours.  How 
long  would  it  take  the  son  to  hoe  the  field  ? 

44.  A  can  do  a  piece  of  work  in  4  hours,  B  can  do  it  in 

6  hours,  and  C  in  3  hours.     How  long  would  it  take  them 
working  together  ? 

45.  A  can  mow  a  field  in  6  hours,  B  in  8  hours,  and  with 
the  help  of  C  they  can  do  it  in  2  hours.  How  long  would 
it  take  C  working  alone  ? 

46.  A  tank  can  be  emptied  by  two  pipes  in  5  hours  and 

7  hours   respectively.      In  what  time  can  it  be  emptied 
by  the  two  pipes  together  ? 

47.  A  cistern  can  be  filled  by  two  pipes  in  4  hours  and 
6  hours  respectively,  and  can  be  emptied  by  a  third  in  15 
hours.  In  what  time  could  the  cistern  be  filled  if  all  three 
pipes  were  running  ? 


130  A   FIRST  BOOK  IN  ALGEBRA. 

48.  A  and  B  together  can  do  a  piece  of  work  in  8  days, 
A  and  C  together  in  10  days,  and  A  by  himself  in  12  days. 
In  what  time  can  B  and  C  do  it  ?  In  what  time  can  A,  B, 
and  C  together  do  it  ? 

49.  John  and  Henry  can  together  paint  a  fence  in  2 
hours,  John  and  Lewis  together  in  4  hours,  and  John  by 
himself  in  6  hours.  In  what  time  can  the  three  together 
do  the  painting  ? 

50.  C  can  do  a  piece  of  work  in  a  days,  and  D  can  do  the 
same  work  in  b  days.  In  how  many  days  can  they  do  it 
working  together  ? 

61.  James  and  Thomas  can  do  a  piece  of  work  in  d  days 
and  James  alone  can  do  it  in  c  days.  How  long  would  it 
take  Thomas  alone  ? 

52.  Solve  3  +  x_l  +  »_2+^^j_ 

3  — a;     1  — a;     2  —  x 

53.  Find  the  value   of + ^ 

{x  -y){x-  z)      (y  -  x)  {y  -  z) 

z^ 
(2;  -x){z-y) 

54.  Expand  (ic' -  2)  (x" -{- 2)  (x""  +  S)  (x^  -  S) . 

55.  Factor  9 a^ -\- 12 ab -\- W,  12  +  7a;H-a;^  ac-2bc 
—  3ad  +  66d 

Illustrative  Example.  At  what  time  between  1  o'clock 
and  2  o'clock  are  the  hands  of  a  clock  (1)  together? 
(2)  at  right  angles  ?    (3)  opposite  to  each  other  ? 


EQUATIONS. 


131 


How  far  does  the  hour  hand  move  while  the  minute 
hand  goes  around  the  whole  circle?  How  far  while  the 
minute  hand  goes  half  around?  What  part  of  the  distance 
that  the  minute  hand  moves  in  a  given  time  does  the  hour 
hand  move  in  the  same  time? 


Fio.  1. 


Fio.  2. 


Fio.  8. 


(1)  Let  ^3/ and  AH  in  all  the  figures  denote  the  posi- 
tions of  the  minute  and  hour  hands  at  1  o'clock,  and  AX 
(Fig.  1)  the  position  of  both  hands  when  together. 

Let  X  =  number  of  minute  spaces  in  arc  MX. 

MX^MH-\^UX. 

x—^-\-^'    Solution  gives  a;  =  5^. 

Hence,  the  time  is  5^  minutes  past  1  o'clock. 

(2)  Let  AX  and  AB  (Fig.  2)  denote  the  positions  of 
the  minute  and  hour  hands  when  at  right  angles. 

Let         X  =  number  of  minute  spaces  in  arc  MBX. 
MBX  =  MH  +HB+  BX. 

a  =  5  -f  :^  + 15.    Solution  gives  x  =  21^. 
Hence,  the  time  is  21-^  minutes  past  1  o'clock. 


132  A    FIRST  BOOK  IN  ALGEBRA. 

(3)  Let  AX  and  AB  (Fig.  3)  denote  the  positions  of 
the  minute  and  hour  hands  when  .opposite. 

Let         X  =  number  of  minute  spaces  in  arc  MBX. 
MBX=MH+  HB  +  BX. 

a;  =  5  +  —  +  30.    Solution  gives  x  =  SSj\. 
Hence,  the  time  is  38  j^  minutes  past  1  o'clock. 

56.  At  what  time  are  the  hands  of  a  clock  together 
between  2  and  3  ?     Between  5  and  6  ?     Between  9  and  10  ? 

57.  At  what  time  are  the  hands  of  a  clock  at  right 
angles  between  2  and  3?  Between  4  and  5?  Between  7 
and  8? 

58.  At  what  time  are  the  hands  of  a  clock  opposite  each 
other  between  3  and  4  ?  Between  8  and  9  ?  Between  12 
andl? 

59.  At  what  times  between  4  and  5  o'clock  are  the  hands 
of  a  watch  ten  minutes  apart? 

60.  At  what  time  between  8  and  9  o'clock  are  the 
hands  of  a  watch  25  minutes  apart  ? 

61.  At  what  time  between  5  and  6  o'clock  is  the  minute 
hand  three  minutes  ahead  of  the  hour  hand  ? 

62.  It  was  between  12  and  1  o'clock;  but  a  man,  mis- 
taking the  hour  hand  for  the  minute  hand,  thought  that 
it  was  55  minutes  later  than  it  really  was.  What  time 
was  it? 

63.  At  what  time  between  11  and  12  o'clock  are  the 
hands  two  minutes  apart? 


EQUATIONS.  133 

Illustrative  Example.  A  courier  who  travels  at  the  rate 
of  6  miles  an  hour  is  followed,  6  hours  later,  by  another 
who  travels  at  the  rate  of  8J  miles  an  hour.  In  how  many 
hours  will  the  second  overtake  the  first  ? 

Let  X  =  number  of  hours  the  second  is  traveling. 

a;  +  5  =  number  of  hours  the  first  is  traveling. 
%\x  —  distance  the  second  travels. 
6(a;  H-  5)  =  distance  the  first  travels. 

8  J  a;  =  6  (a;  +  5) .     Solution  gives  x  =  12. 
He  will  overtake  the  first  in  12  hours. 

64.  A  messenger  who  travels  at  the  rate  of  10  miles  an 
hour  is  followed,  4  hours  later,  by  another  who  travels  at 
the  rate  of  12  miles  an  hour.  How  long  will  it  take  the 
second  to  overtake  the  first  ? 

65 .  A  courier  who  travels  at  the  rate  of  19  miles  in  4  hours 
is  followed,  8  hours  later,  by  another  who  travels  at  the  rate 
of  19  miles  in  3  hours.  In  what  time  will  the  second 
overtake  the  first?  How  far  will  the  first  have  gone 
before  he  is  overtaken  ? 

66.  A  train  going  at  the  rate  of  20  miles  an  hour  is 
followed,  on  a  parallel  track,  4  hours  later,  by  an  express 
train.  The  express  overtakes  the  first  train  in  5^  hours. 
What  is  the  rate  of  the  express  train  ? 

67.  A  messenger  started  for  Washington  at  the  rate  of 
6J  miles  an  hour.  Six  hours  later  a  second  messenger 
followed  and  in  4rJ  hours  overtook  the  first  just  as  he  was 
entering  the  city.  At  what  rate  did  the  second  messenger 
go  ?     How  far  was  it  to  Washington  ? 


134  A    FIRST  BOOK  IN  ALGEBRA. 

68.  How  far  could  a  man  ride  at  the  rate  of  8  miles  an 
hour  so  as  to  walk  back  at  the  rate  of  4  miles  an  hour  and 
be  gone  only  9  hours  ? 

69.  Two  persons  start  at  10  a.m.  from  towns  A  and  B, 
55|-  miles  apart.  The  one  starting  from  A  walks  at  the 
rate  of  4J  miles  an  hour,  but  stops  2  hours  on  the  way ; 
the  other  walks  at  the  rate  of  3J  miles  an  hour  without 
stopping.  When  will  they  meet?  How  far  will  each 
have  traveled? 

Suggestion.    Let  x  equal  the  number  of  hours. 

70.  A  boy  who  runs  at  the  rate  of  12^  yards  per  second, 
starts  16  yards  behind  another  whose  rate  is  11  yards  per 
second.  How  soon  will  the  first  boy  be  8  yards  ahead  of 
the  second? 

71.  A  rectangle  whose  length  is  4  ft.  more  than  its 
width  would  have  its  area  increased  56  sq.  ft.  if  its  length 
and  width  were  each  made  2  ft.  more.  What  are  its 
dimensions  ? 

72.  The  length  of  a  room  is  double  its  width.  If  the 
length  were  3  ft.  less  and  the  width  3  ft.  more,  the  area 
would  be  increased  27  sq.  ft.  Find  the  dimensions  of  the 
room. 

73.  A  floor  is  two-thirds  as  wide  as  it  is  long.  If  the 
width  were  2  ft.  more  and  the  length  4  ft.  less,  the  area 
would  be  diminished  22  sq.  ft.     What  are  its  dimensions  ? 

74.  A  rectangle  has  its  length  and  width  respectively 
4  ft.  longer  and  2  ft.  shorter  than  the  side  of  an  equivalent 
square.     Find  its  area. 


EQUATIONS.  186 

75.  An  enclosed  garden  is  24  ft.  greater  in  length  than  in 
width.  684  sq.  ft.  is  used  for  a  walk  3  ft.  wide  extending 
around  the  garden  inside  the  fence.  How  long  is  the 
garden? 

76.  Factor  ^-^'-,»    ^-27f,     a"-6^     2c-4c^-f-2c«. 

4      9m^ 

77.  Extract  the  square  root  of  12a^-24x-i-9-{-x'^-22a^ 
-4x'-{-2Sa^. 

78.  What  must  be  subtracted  from  the  sum  of  4ar'4-3iB*2/ 
—  2/^,  ^oc^y  —  Sx^,  7x^y  -\-9i^  —  2  xh/,  to  leave  the  remainder 
2x^-30^  +  ^? 

79.  Find  the  G.  C.  F.  of  a;(a;4-l)',  x'ix'-l),  and 
2a:»-2a^-4x. 

80.  From  one  end  of  a  line  I  cut  off  5  feet  less  than 
one-fifth  of  it,  and  from  the  other  end  4  feet  more  than  one- 
fourth  of  it,  and  then  there  remained  34  feet.  How  long 
was  the  line  ? 

81.  A  can  do  twice  as  much  work  as  B,  B  can  do  twice 
as  much  as  C,  and  together  they  can  complete  a  piece  of 
work  in  4  days.  In  what  time  can  each  alone  complete  the 
work. 

82.  Separate  57  into  two  parts,  such  that  one  divided  by 
the  otlier  may  give  5  as  a  quotient,  with  3  as  a  remainder. 

83.  Divide  92  into  two  parts,  sucli  that  one  divided  by 
the  other  may  give  4  as  a  quotient,  with  2  as  a  remainder. 


136  A   FIRST  BOOK  IN  ALGEBRA. 

84.  Fourteen  persons  engaged  a  yacht,  but  before  sail- 
ing, four  of  the  company  withdrew,  by  which  the  expense 
of  each  was  increased  $  4.     What  was  paid  for  the  yacht  ? 

85.  Find  two  consecutive  numbers  such  that  a  fifth  of 
the  larger  shall  equal  the  difference  between  a  third  and  an 
eighth  of  the  smaller. 

86.  A  is  24  years  older  than  B,  and  A's  age  is  as  much 
above  50  as  B's  is  below  40.     What  is  the  age  of  each  ? 

87.  Find  the  number,  whose  double  added  to  16  will 
be  as  much  above  70  as  the  number  itself  is  below  60. 

88.  A  hare  takes  5  leaps  to  a  dog's  4,  but  3  of  the  dog's 
leaps  are  equal  to  4  of  the  hare's ;  the  hare  has  a  start  of 
20  leaps.  How  many  leaps  will  the  hare  take  before  he  is 
caught  ? 

Suggestion.  Let  bx  equal  the  number  of  leaps  the  hare  will  take, 
and  let  m  equal  the  length  of  one  leap. 

89.  A  greyhound  takes  3  leaps  to  a  hare's  5,  but  2  of 
the  greyhound's  leaps  are  equal  to  4  of  the  hare's.  If  the 
hare  has  a  start  of  48  leaps,  how  soon  will  the  greyhound 
overtake  him  ? 

90.  A  hare  has  40  leaps  the  start  of  a  dog.  When  will 
he  be  caught  if  5  of  his  leaps  are  equal  to  4  of  the  dog's, 
and  if  he  takes  7  leaps  while  the  dog  takes  6  ? 


EQUATIONS.  137 


SIMULTANEOUS    EQUATIONS. 

x  -f  w  =  8,        x  =  3}  .    ^  ^, 

„  ^  r  in  both  equations. 

x-y  =  7.       y  =  5) 


48.   Illus.     x-]-y  =  S,        a;  =  3^. 
4 


Simultaneous  equatious  are  equations  in  which  the 
same  unknown  numbera  have  the  same  value. 

One  equation  containing  more  than  one  unknown 
number  cannot  be  solved.  There  must  be  as  many 
simultaneous  equations  as  there  are  unknown  numbers. 

ILLUS.  1.     Solve  1/  +  '^  =  "' 
\2x-{-    y=    9. 

Multiply  the  first  equation  by  2 ; 

then  2x4-6^  =  34 

but  .      2a;H-    y=   9 

Subtracting,  5y  =  25 

y=    5. 

To  find  the  value  of  «,  substitute  the  value  of  y  in  the 
second  equation : 

2x-f  5  =  9,   2x  =  4,   a;  =  2. 


ILLU8.2.    Solve  f/  +  '^  =  '^' 
\bx-Qy=    1. 


138  A    FIRST  BOOK  IN  ALGEBRA. 

Multiply  the  first  equation  by  3,  and  the  second  equa- 
tion by  2, 

9a; +  122/ =  36 

10a; -122/=    2 
Adding,  19a;  =38     .-.a;  =  2. 

Substituting,     6 +  4?/=  12,   4?/ =6,   y  =  l^. 

Multiply  one  or  both  of  the  equations  by  such  a 
nuinber  that  one  of  the  iinhnown  numbers  shall 
have  lihe  coefficients.  If  the  signs  of  the  terms 
having  lihe  coefficients  are  alihe,  subtract  one  equa- 
tion from  the  other ;  if  unlihe,  add  the  equations. 

Exercise  55. 

Solve : 

^^    I      x+    2/  =  4,  g     j    2a;+   92/  =  -5, 


2. 


4. 


I    3a; -22/ =  7.  *    \llx  ■\-Wy  =  l. 

(      X-    y  =  2,  g      (    42/-  2a;  =  4, 

(   2a;  +  52/=18.  *    t'102/+  3a;  =  -8. 

^      (    5a;  +  22/  =  47,  lo     I    ^''~  ^^^^^' 

^'    I   2x-    2/  =  8.  '    ^    5^-+  32/=8. 

.    ^         o        -<A  (    ^y—  2a;  =  3, 

(    4a;  — 3?/ =  10,  11.    -^ 

]    n     ,   /      ,a  ^    4v-  6a;  =  2i 

(   6a;  4- 4?/ =  49.  "^                   '^ 

(    3a;  +  2y=ll, 

8a;-22/  =  6,  12.    -^    _  /      .   ' 

•^        '  (7a;—  5?/=  190. 
10aj  +  72/  =  36. 

^      (    2a;-52y  =  -ll,  13.    j    ^^^  |2/=102. 

'    ^    ^•'^'+    ^=^'  .    5a;+  22/  =  66, 

I    7.-32/  =  41,  14.    -        .  3,^ 

1    2a;-    2/  =  12.  (34' 


EQUATIONS. 


139 


15. 


16. 


5        7 
a;-|-2i/  =  -63. 


2a; 


+  73/ =189. 


^±1^=1 

17.  ^      /^-^ 

13  +  05-22/       ^* 

18.  ^-^ 
2y-4x^      ^ 

L     3-7/ 


2y  +  a;__2a;  +  y 


8i, 


19. 


20. 


21. 


22. 


3  4 

3a;  +  y  _  .V  ^  109     4y  — a; 
2  3  ~  10  5     * 

^  a;  +  2/  =  a, 
(x  — y=  6. 

3a; -19  ,  ^_3y  +  a;  ,  5a? -3 

4a;+5y     2a;  +  y_9a;  — 7      3y  +  9 
16  -2      ~      8  4     ' 

i  (3a; -22/) +  i  (5a; -32/)=  a;, 

l£^+Ja;-2/  =  l+'2/. 


23.  If  1  is  added  to  the  numerator  of  a  fraction,  its 
value  is  J;  but  if  4  is  added  to  its  denominator,  its  value 
is  \.     What  is  the  fraction  ? 

Suggestion.  Lettiug  x  equal  the  numerator,  and  y  the  denomi- 
nator, form  two  equations. 

24.  If  2  is  subtracted  from  both  numerator  and  denomi- 
nator of  a  certain   fraction,  its  value   is  J;    and   if  1  is 


140  A   FIRST  BOOK  IN  ALGEBRA. 

added  to  both  numerator  and  denominator,  its  value  is  f . 
Wliat  is  the  fraction  ? 

25.  If  2  is  added  to  both  numerator  and  denominator  of 
a  certain  fraction,  its  value  is  f ;  but  if  3  is  subtracted 
from  both  numerator  and  denominator,  its  value  is  ^. 
What  is  the  fraction  ? 

26.  If  3  be  subtracted  from  the  numerator  of  a  certain 
fraction,  and  3  be  added  to  the  denominator,  its  value  will 
be  i ;  but  if  5  be  added  to  the  numerator,  and  5  be  sub- 
tracted from  its  denominator,  its  value  will  be  2.  What  is 
the  fraction? 

27.  The  sum  of  two  numbers  divided  by  2  is  43,  and 
their  difference  divided  by  2  is  19.    What  are  the  numbers  ? 

28.  The  sum  of  two  numbers  divided  by  3  gives  as  a 
quotient  30,  and  their  difference  divided  by  9  gives  4. 
What  are  the  numbers  ? 

29.  Five  years  ago  the  age  of  a  father  was  four  times 
that  of  his  son ;  five  years  hence  the  age  of  the  father  will 
be  2^  times  that  of  the  son.     What  are  their  ages  ? 

30.  Seven  years  ago  John  was  one-half  as  old  as  Henry, 
but  five  years  hence  he  will  be  three-quarters  as  old. 
How  old  is  each  ? 

31.  A  and  B  own  herds  of  cows.  If  A  should  sell  6 
cows,  and  B  should  buy  6,  they  would  have  the  same  num- 
ber; if  B  should  sell  4  cows  to  A,  he  would  have  only 
half  as  many  as  A.  How  many  cows  are  there  in  each 
herd? 


EQUATIONS.  141 

32.  The  cost  of  5  pounds  of  tea  and  7  pounds  of  coffee  is 
1^4.94  j  the  cost  of  3  pounds  of  tea  and  6  pounds  of  coffee 
is  $3.54.  What  is  the  cost  of  the  tea  and  coffee  per 
pound? 

33.  What  is  the  price  of  com  and  oats  when  4  bushels 
of  com  with  6  bushels  of  oats  cost  $4.66,  and  5  bushels  of 
com  with  9  bushels  of  oats  cost  $6.38  ? 

34.  A  merchant  mixes  tea  which  cost  him  87  cents  a 
pound  with  tea  which  cost  him  29  cents  a  pound.  The 
cost  of  the  mixture  is  $17.98.  He  sells  the  mixture  at 
55  cents  a  pound  and  gains  $2.92.  How  many  pounds  of 
each  did  he  put  into  the  mixture  ? 


»:•:< 


QUADRATIC   EQUATIONS. 

49.   Illus.  1.     ax'  =  b,     7x2- 10  =  5 +  2a^. 

Illus.  2.     a^  +  8x  =  20,     ax^ -{- hx  -  c  =  ba?  ■\- d. 

A  quadratic  equation  is  an  equation  in  which 
the  highest  power  of  the  unknown  number  is  a  square. 
It  is  called  an  equation  of  the  second  degree. 

If  it  contains  only  the  second  power  of  the  unknown 
number  (Illus.  1),  it  is  called  a  pure  quadratic  equa- 
tioo.  If  it  contains  both  the  first  and  second  powei*8  of 
the  unknown  number  (Illus.  2),  it  is  called  an  affected 
quadratic  equation. 


142  A    FIRST  BOOK  IN  ALGEBRA. 

50.  iLLus.      Solve    ^._^-'-10^35_^  +  50. 

3  5 

IBx"  -  5a^  -\-  50  =  525  -  3x^  - 150. 
13a;2  =  325. 

a^  =  25. 

a;  =  ±  5. 

To  5oZz;6  <x  pure  quadratic  equation,  reduce  to  the 
form  x^  =  a  and  take  the  square  root  of  each  member. 

Exercise  56. 

Solve : 

1.  5a;2-12  =  33.  2         14 

5.    — ^  =  ^. 

2.  3ar  +  4  =  16.  ^^     ^^'     ^^ 

3.  40:^  +  11=136-0.1  ^     t^zl^l==t±l. 

4.  5(3a^-l)  =  ll(o;2  +  l).        *       6         4         8 

7.  (o;  +  3)2  =  6o;  +  58. 

8.  6o.  +  24-^  =  i^±^-li. 

a;  4 

9.  £1:1^  +  ^  +  1+ § =  0. 

x-1      x  +  3      x^  +  2x-'d 

10     fl?  +  l  ■  2(o;-3)^     16 -9o; 
*o;-l         o;-2     ""o;2-3o;  +  2* 

12  1 

11-   -7^ -. T  —  —. ^  + 


(2_o;)(3-o;)       (l-oj)(a;-3)       (o;-l)(o;-2) 

1  +  1 


2-x      {x-l){2-x){x-3) 


12.    i^_   i_ .    10 


6a; +  6     2a;  +  2     S-Sa;^     3(1 -a;) 


EQUATIONS.  143 

13.  A  father  is  30  years  old,  and  his  son  is  two  years 
old.  In  how  many  years  will  the  father  be  three  times  as 
old  as  his  son  ? 

14.  Divide  the  number  112  into  two  parts  such  that  the 
smaller  divided  by  their  difference  will  give  as  a  quo- 
tient 3. 

15.  The  numerator  of  a  fraction  is  4  less  than  the  denom- 
inator ;  if  30  be  added  to  the  denominator,  or  if  10  be  sub- 
tracted from  the  numerator,  the  resulting  fractions  will 
be  equal.     What  is  the  original  fraction  ? 

1-1       T  01  35  —  1        X  —  3  2 

51.   Illus.     Solve -  =  — -. 

x-2     x-4:        3 

3a^  -  15x  +  12  -  3»2  ^  1535  _  18  ^  _  2  ar  +  12aj  - 16. 
2ar'-12a;  =  -10. 
x^  —  6x  =  —5. 
a:2_6a;  +  5  =  0. 
(x-5){x-l)  =  0. 

This  equation  will  be  satisfied  if  either  factor  is  equal 

to  zero.     Placing  each  factor  in  turn  equal  to  zero,  and 

solving, 

a;-5  =  0,  a;-l=0, 

x  =  5'j  x=l. 

Ans.  aj  =  5  or  1. 

To  solve  an  affected  quadratic  ^  equation,  reduce 
the  equation  to  the  form  x'^  +  bx  +  c  =  0,  factor  the 
first  meTyiber,  place  each  factor  in  turn  equal  to 
zero,  and  solve  tJie  simple  equations  thus  formed. 


144  A    FIRST  BOOK  IN  ALGEBRA. 

Exercise  67. 

Solve : 

1.  a;2  +  3a;  =  18.  6.  lS7  =  x'  +  6x. 

2.  x^-{-5x=U.  7.  x'-2bx  =  -b\ 

3.  a;(a;-l)  =  72.  8.  x'  =  4.ax-Za\ 

4.  a;2  =  10x-21.  9.  x^ -\-{a-l)x  =  a. 

5.  23a;  =  120-|-aj2.  10.  adx  —  aGX^=hcx  —  bd. 

11.  (a;  +  3)(x-3)  =  8(a;  +  3). 

12.  (a;  +  2)(ic-5)  =  4(a;-4). 

13.  ^  +  ?  =  12  16.    _-^_2i  +  ^^^  =  0. 
5      a;  x  +  l  X 


X        ct         X  X 


24 


14.    :^-2  =  — --.  17.    a;4-4  =  3a; 

3  12      2  aj-1 

_         8         o  ,  ^  +  l_o       lo     a?-l      a;  +  3_2(a;  +  2) 


19. 


x-l      3x^4-2^    3a; 
x-^2       a;2-4       2- a;' 


2a;        2a;(a;-3)  ^  a;-3 
'3-a;         a;2_9         ^_^3' 

21.  At  what  time  between  4  and  5  o'clock  are  the  hands 
of  a  clock  opposite  each  other  ? 

22.  John,  having  three  times  as  much  money  as  Lewis, 
gave  Lewis  $2,  and  then  had  twice  as  much  as  Lewis. 
How  much  had  each  at  first  ? 

23.  A  fish  is  3  feet  long;  its  head  is  equal  in  length  to 
the  tail,  and  its  body  is  five  times  the  length  of  the  head 
and  tail  together.     What  is  the  length  of  the  head  ? 


EQUATIONS.  145 

24.  In  how  many  days  can  A,  B,  and  C  build  a  boat 
if  they  work  together,  provided  A  alone  can  build  it  in 
24  days,  B  in  18  days,  and  C  in  30  days  ? 

The  above  method  of  solving  affected  quadratic  equations 
is  the  simplest  of  three  methods  commonly  used,  and  will 
not  solve  all  possible  cases ;  the  method  given  for  solving 
simultaneous  equations  is  only  one  of.  three  known  meth- 
ods ;  the  cases  in  factoring  are  less  than  half  of  those 
usually  taken.  In  fact,  we  have  made  only  a  beginning  in 
the  subject  of  algebra;  much  more  lies  ahead  along  the 
lines  which  we  have  been  following.  Can  you  grasp 
more  clearly  the  conditions  given  in  any  problem 
presented  to  you,  and  see  more  definitely  Just  what 
is  required,  than  when  you  began  this  study?  Do 
you  possess  greater  ability  to  think  out  problems? 
Has  the  use  of  letters  to  represent  numbers  made 
you  think  more  exactly  what  is  to  be  done,  and 
what  the  operations  mean?  If  so,  your  knowledge  of 
numbers  is  broader,  and  you  already  know  that 

Algebra  is  the  knowletlge  which  has  for  its  object 
general  truths  about  numbers. 

Exercise  68.     (General  Review.) 

I. 
1.    When   a=l,   6  =  3,   c  =  Of   and  d  =  0,    what  is  the 
value  of 

4a+b^+b^c^-\-ad       l+a'c^        g^+b^^ct'      a^-h2a6+6V 


146  A    FIRST  BOOK  IN  ALGEBRA. 

2 .  Prove  that    (x^  -{-xy-{-  if)  (x-  —  xy  +  y')  =  ^^  ~  ^^ 

x'^  —  y^ 

3.  Solve  {x-{-5y-(x-\-iy-16x  =  {x-iy-{x- 5)2. 

4.  A  tank  is  filled  by  two  pipes,  A  and  B,  running 
together,  in  12  hours,  and  by  the  pipe  B  alone  in  20  hours. 
In  what  time  will  the  pipe  A  alone  fill  it? 

5.  Find  the  G.  C.  F.  of  x^-{-l-x-x^,  aP-\-x-l-a^, 
x^  —  1,   and   x'^  —  4:X  +  3. 

6.  Divide  a'  -f  a*b  -  a^b^  +  o?  -  2ah^  +  ¥  by  o?-h-{-  a. 

7.  Find  the  square  root  of  6y^ +  1 -{-V^jf  —  2y —  2'i^ 
H- 4/ +  72/2. 

8.  Expand 
(a;+l)(x-+2)-(2a;+l)(2a;-h3)  +  (a;-4)(a;-9)  +  (a;-5)^. 

9.  Solve  ?^  +  ^I^  =  l 

x-2     x+2     2 

10.  Simplify 
-1      {x-\-3){x-l)J      \x^3      {x-l){x^3)) 

II. 

11.  Add  2(a-c)3-10a^.?/-7(tt-c),  6(a-c)-2(a-c)^ 
-lOT^y,  3(a-c)-(a-c)3-f-2a^2/j  2{a-c)-^a?y -{a-cf, 
4(a -c)  +  5{a-cY-i- 2x'y,   3(a -c)-2o?y -Q{a- cy. 

12.  Solve   hx-l)'=^3h''-Ux. 

13.  Factor  ic«+2a^-3,  ax'-ay'+by^-bx^,  27x^+{y+zy. 


EQUATIONS.  147 

14.  Find  the  fraction  which  becomes  equal  to  one  when 
six  is  added  to  the  numerator,  and  equal  to  one-third  when 
four  is  added  to  the  denominator. 

h^     a 

15.  Simplify 


fr^      b''     ab 


16.    Solve   I^  =  (x-?^)  +  7. 


17.  Six  years  ago  John  was  five  times  as  old  as  Sarah. 
If  he  is  twice  as  old  as  Sarah  now,  what  are  their  ages  ? 

18.  Multiply  together  \^^,   ^— ^»  and  1  +  — ^• 

1+2/     .T  +  ar  1  —  x 

19.  Simplify 

20.  X  times  y  is  how  many  times  a  ? 

III. 

21.  Add  2x-{-y-2a-\-55^b,  24&-y  +  2a;-|- a,  3a 
—  2y  —  4x  —  Slb,  and  subtract  the  result  from  2y-^3a 
+  i6  4-3x. 

22.  Divide   i^a*  -  ^a^  — ^a-^a*  by   a^-^a. 

23.  A  can  do  a  piece  of  work  in  3  days  which  B  can  do 
in  5  days.     In  what  time  can  they  do  it  working  together  ? 

24.  Simplify   ---«-*_  +-^,  +  ^. 

a'  — oft  +  y     o'  +  fr'     0  +  6 


148  A   FIRST  BOOK  IN  ALGEBRA. 

25.  Factor  a;2-9a;-52,  1  -  a^  (a^ -\- by -\- 2 (a' -  b*) 
^(^a'-by. 

26.  (Solve ^  =  x  —  6 

4  2  2 

27.  The  sum  of  the  ages  of  a  man  and  his  son  is  100 
years ;  one-tenth  of  the  product  of  their  ages  exceeds  the 
father's  age  by  180.     How  old  are  they  ? 

28.  Solve    iB=9-^,   y  =  ll-^^. 

29.  From  what  must  3a;''  — 2a^4-£C  — 6  be  subtracted 
to  produce  unity  ? 


30.  Find  the  following  roots  :  V5.5225,    V32.768. 

IV. 

31.  Find  the  value  of  4^?  +  V ^  2£+^^  _  5^^ 
if  x=l,   y  =  2,   2  =  0,   a  =  4,   and   6  =  5. 

32.  Solve    — ?_  =  _i_. 

x  —  Q      a  — 9      x—'S 

33.  Find  three  consecutive  numbers  whose  sum  is  78. 

34.  Find  the  G.  C.  F.  of  2o?-12a-2a\  a' -  ^a\  and 
4a»6  +  16a6  +  16a26. 

35.  Divide  — ^^~y'     ,  by  t±M. 

a?  —  2xy  -\-y^  X  —  y 

36.  A  fraction  becomes  |  by  the  addition  of  three  to  the 
numerator  and  one  to  the  denominator.  If  one  is  subtracted 
from  the  numerator  and  three  from  the  denominator,  it  be- 
comes \.     What  is  the  fraction  ? 


EQUATIONS.  149 

37     Expand   f     3°'ft(m  +  n)V    J50=^(a+3L 

38.  If   a,  certain   number  is   multiplied   by   itself,   the 
result  is   9  a;*  —  4  a;  +  10  a^  +  1  —  12  ar*.     Find  the  number. 

o«     o-      ^•c       ax  — or        a^-{-ax         2  ax 

39.  S.mphfy    __,  x  ^^,  ^  ^,-^. 

40.  Solve  182-202^  =  3,  il^-^  =  0. 

V. 

41.  Factor   x"-}- 5x^  +  6,     x'-Ux-\-49y     x^-{y+zy. 

42.  Add   xy-^gx-j\{x'-y')-5x^fj     ^x-xy-\-9a^y^ 
-hii^-f),     i^f-xy-hix  +  iix^-f),     2xy  +  ix 

43.  At  what  times   between  7  and   8   o'clock   are   the 
hands  of  a  clock  six  minutes  apart  ? 

AA     c-      vf     x'-Bx  +  G  ^x'-^x+S     a^-6xH-9 
45.    Solve   ^^^  =  2- 


6+2  6+1 

46.  Factor  t-%    ^-^^-14,    ^-2  +  g. 

%f       (?      f      y  f  aj2 

47.  A,  who  works  only  two-thirds  as  fast  as  B,  can 
build  a  stone  wall  in  12  days.  In  what  time  could  A 
and  B  together  build  the  wall  ? 


150  A   FIRST  BOOK  IN  ALGEBRA. 


48.  Solve  ^±^-^^1^  =  8,    ^1^  +  ^:^  =  11. 

49.  Expand    {l-{-2xy,     {2x'-3a'by. 

50.  Eeduce    (a^ +  2a-^-  + 6-)(a- +  6^)    to  lowest  terms. 

a^  —  b^ 


VI. 

51.  y  is  how  mucli  greater  than  x? 

52 .  Subtract  3 oc^  -\-  4:X^y  —  7  xy'^  -^  10 y^  from  4:X^  —  2x-y 
-i- 4:xy^ -\- Ay^  and  find  the  value  of  the  remainder  when 
x  =  2   and  y  =  1. 

53.  The  length  and  width  of  a  rectangle  are  respectively 
5  feet  longer  and  4  feet  shorter  than  the  side  of  an  equiv- 
alent square.     What  is  its  area  ? 

54.  Find  the  L.C.  M.  of  a^- 3- 2a,  a^-l,  and  2a^ 
-6a +  4. 

A_i4.« 

«.      ,.„     4a  b 

55.  Simplify      _^  _  ^  » 

2a       6 

56.  Solve    ^^  +  _A_  =  2. 

3         ic  —  1 

57 .  Factor  a^  6  +  8  ac^dm^    4  c^^  +  03/^  +  4 c^a;^/,     a?^  -  1. 

58.  Multiply  l-^x-^x^-^^x^  hj  l-^x^-^a^-^x. 

59.  Find  the  cube  root  of  6a;^+7ar^+3aj^+6a;H«*^+l  +  3a;. 

60.  Divide    12 oj^i/^  _  4 2/'  -  6 a;^?/  +  x'  hj  a^ +  2y'' -S xy. 


EQUATIONS.  161 

VII. 

61.  Add  3i^a«_|a*_ia«  +  Aa,  \a' -ia-^a'-\c?, 
^a^  +  ia^^  +  ia^  +  fa,     fa'  +  ta^  +  ia^  +  TVa. 

62.  Solve   x{a  —  x)-{-x{h  —  x)  =  2{x  —  a){h  —  x). 

63.  Factor  a;* - 22 a?* - 75,     IQ-^,     («+ 6)2_(a-6)«. 

64.  A  piece  of  work  can  be  finished  by  3  men  in  8  days, 
or  by  5  women  in  6  days,  or  by  6  boys  in  6  days.  In  what 
time  can  2  men,  3  women,  and  3  boys  do  the  work  ? 

65.  Solve  3^-(^l  +  3)=55±2_^3_3^1^. 

67.  What  number  is  that,  the  sum  of  whose  third  and 
fourth  parts  is  less  by  two  than  the  square  of  its  sixth  part  ? 

68.  Solve    1-1  =  1,    f -1  =  3. 

69.  Divide  m  hy  l-\-y  to  four  terms. 

70.  If  a:  is  {  of  a  number,  what  is  the  number? 

VIII. 

71.  The  head  of  a  fish  is  6  inches  long,  the  tail  is 
as  long  as  the  head  and  half  the  body,  and  the  body  is 
as  long  as  the  head  and  tail.  What  is  the  length  of  the 
fish? 

72.  Add  4a  — 5x  — 152/,  a-flSx  +  Sy,  4a  — 7a;+lly, 
a+3a;4-5y,  and  multiply  the  result  by  the  difference 
between  11a +  7y  and   10aH-6y  — a;. 


152  A   FIRST  BOOK  IN  ALGEBRA. 

73.  Divide  20524.  |ic4_|_  I  by  2a; +  3a^+|. 

74.  How  many  numbers  each  equal  to  l  —  2x-{-x^  must 
be  added  together  to  equal  5x^  —  6x^-^1? 

75.  Factor  a^ +5 a' -4. a -20,     afi-y^,    2a^-Sa^y'+6xy\ 

.  76.  A  courier  who  travels  at  the  rate  of  5  miles  an  hour 
is  followed,  4  hours  later,  by  another  who  travels  at  the  rate 
of  15  miles  in  2  hours.  In  how  many  hours  will  the 
second  overtake  the  first  ? 

77.  Divide   — ^  by  ^— +^_. 

1-x      l-\-x     -^   1-x     l-\-x 

78.  Solve  3 a; -42/ =-6,     10a; +  2?/ =  26. 

79.  Scby  -  3a^  +  U^  -  5cd  +  4xy  -  6a^  -7 b^-\-7 cd-\-3xy 
-  6a^  -^  6b'  -  3cd  -  5xy  -{-  7 a"  -  6b^  +  4cd  -{-  ^xy  -^Ta' 
-7b'  +  4.cd-6xy-6a'-\-3b'-7cd  +  7a'=? 

80.  Simplify 

3  a;  -  5  -  J  2  ( 4  -  a;)  -  3  ( a;  -  2 )  j -f- 5  3  -  ( 5  +  2  a;)  -  2 1 . 


ANSWERS 


TO 


A   FIRST   BOOK   IN   ALGEBRA. 


Exercise  1. 

1.  43;  86.  10.   Lot,  $  720  ;  house,  ^  3600. 

2.  Carriage,  $376 ;  horse,  $125.     11.    Mr.  A,  72  ;  son,  24. 

3.  C,  $31 ;  J,  $156.  12.    50  A. ;  300  A. 

4.  8  ;  56.  13.    Diet.,  $7.20  ;  rhet,  $.90. 
6.  8  miles.                                         14.    112  ;  4144. 

6.  Needles,  8f ;  thread,  64<?.  15.   Aleck,  56.<? ;  Arthur,  Sf. 

7.  224  giris  ;  448  boys.  16.    Mother,  28  ;  daughter,  4. 

8.  25 ;  275.  17.   J,  15  yrs. ;  M,  6  yrs. 

9.  H,  6  qts. ;  J,  18  qts. 

Exercise  2. 

1.  Necktie,  $ .75  ;  hat,  $3;  boots,  $3.75.        2.   30;  46;  16  miles. 

3.  James,  16 ;  sister,  5  ;  brother,  10.  5.  A,  35 ;  B,  16 ;  C,  5. 

4.  Pig,  $  10  ;  cow,  $30 ;  horse,  $60.  6.    12  ;  48. 

7.  8  men  ;  40  women. 

8.  Henry,  $200  ;  John,  $400  ;  James,  $800. 

9.  4500  ft. ;  13,500  ft. ;  27,000  ft.         11.    165  ;  33  ;  11. 

10.  16;  45;  60  pigeons.  12.    A,  $44;   B,  $11  ;  C,  $66. 

13.  Calf,  $8 ;  cow,  $16  ;  horse,  $48.    16.    Cow,  $30  ;  lamb,  $5. 

14.  150  ;  450  gal.  16.    Tea,  dOf  ;  coffee,  30ft. 
17.  Mre.  C,  $26,000;  Henry,  $6000. 


Copyright,  1894,  by  Silvrk,  Bukdktt  akd  Compakt. 
153 


154  A   FIRST  BOOK  IN  ALGEBRA. 

Exercise  3. 

1.  14  boys  ;  21  girls.    .  10.    16  ;  19  ;  21. 

2.  14  yrs. ;  29  yrs.  11.   21 ;  17  ;  24. 

3.  492;  587  votes.  12.    $10,000;  $11,500;  $12,700. 

4.  22  ;  48.  13.    21 ;  38  ;  6. 

5.  J,  79  ;  H,  64.  14.    51 ;  28  ;  16  sheep. 

6.  Flour,  27  bbls. ;  meal,  30  bbls.  15.    A,  253  ;  B,  350  ;  C,  470  votes. 

7.  23  Hoi. ;  40  Jer.  16.    17  ;  12  ;  24  A. 

8.  $18;  $26.  17.   36;  20;  55. 

9.  40;  59.  18.   $50,000;  $44,000;  $24,000. 

Exercise  4. 

1.  C,  34 ;  H,  15.  4.   65.       7.   11. 

2.  26  pear ;  7  apple.  5.   18.       8.   Tea,  $8.76  ;  coffee,  $1.63. 

3.  J,  16  qts. ;  M,  7  qts.  6.   24.       9.    15  ;  33  rooms. 

10.  5;  6;  12.  13.    $50;  $68;  $204. 

11.  17;  20;  100.  14.    A,  $5000;  B,  $10,500;  C,  $31,500. 

12.  $5000;  $3000;  $10,000.  15.    8000;  24,250;  48,500  ft. 

16.  Daughter,  $25,000  ;  son,  $40,000  ;  widow,  $160,000. 

17.  Father,  14  qts. ;  older  son,  7  qts. ;  younger  son,  4  qts. 

18.  H,  200  stamps  ;  J,  185  stamps ;  T,  189  stamps. 

Exercise  5. 

1.  Blue,  5  yds. ;  white,  15  yds.    5.    12. 

2.  3.  6.   12  twos  ;  24  fives. 

3.  Walked  2  hrs. ;  rode  8  hrs.     7.    Tea,  67  f;  coffee,  32)^. 

4.  Book,  $2;  lamp,  $4.  8.   Crackers,  18/*;  gingersnaps,  25)^^. 
9.  Lamp,  $  1 ;  vase,  $  1.50.  14.   27  ;  10  ;  42. 

10.  House,  $4500;  barn,  $3300.  15.   3;  17;  51. 

11.  12,000  ;  13,500  ft.  16.   3  bbls. ;  9  boxes. 

12.  29  gal.;  24  gal.  17.    18;  90;  180. 

13.  Johnson,  $6000;  May,  $1500.  18.   84;  132. 


ANSWERS. 

Exercise  6. 

1. 

4. 

6. 

$8. 

2. 

7. 

6. 

8  sheep. 

3. 

12yr8. 

7. 

121 ;  605. 

4. 

$6.25. 

8. 

142;  994. 

156 


9.  $500;  $1450;  $2900. 

10.  W,  6  yrs.  ;  J,  9  yrs. 

11.  25  ;  15  marbles. 

12.  16. 

13.  Oranges,  35^  ;  apples,  20^.         16.   Cow,  $  30  ;  horse,  $  46. 

14.  9;  15.  17.   5. 

15.  30  yrs. ;  32  yrs.  18.  Boots,  $5  ;  clothes,  $18. 

Exercise  7. 

1.  14  yrs.;  56  yrs.  3.   $3000;  $9000. 

2.  Corn,  60  ;  wheat,  300.  4.   70  mHes  ;  35  miles. 

5.  45;  720.                              9.  $40,000.  13.  90. 

6.  189.                                  10.  24  marbles.  14.  60,000  ft. 

7.  J,  3  yrs. ;  M,  15  yrs.         11.  $30,000.  15.  70  ft. 

8.  96.                                    12.  300  oranges.  16.  72  sq.  rds. 
17.  A,  $22,500;  B,  $7500.                          18.  30. 

Exercise  8. 

1.  4.                                         4.    16.  7.  $3000. 

2.  45  marbles.                        5.  45.  8.  24. 

3.  12  ;  24  ;  6  cows.                 6.    6048.  9.  14. 

10.  48,000  ft.  13.    56  ;  21  ;  7. 

11.  30.  14.    16  ;  4  ;  56  ;  36. 

12.  18 ;  9  ;  63.  15.   Coffee,  18  lbs. ;  tea,  20  lbs. ;  cocoa,  24  lbs. 
16.' $2500;  $6000;  $7500.        17.   J,  9)^ ;  P,  81^  18.   $12,000. 

Exercise  0. 

1.  36 ;  21.  5.  J,  10  boxes  ;  H,  16  boxes.  9.  240  girls;  180  boys. 

2.  24 ;  18.  6.  33  tons  ;  27*  tons.  10.  150  lemons. 

3.  42;  30miles.  7.  John,  28 yrs.;  James,  32 yrs.  11.  21,000  ft. ;  6000  ft. 

4.  30 ;  64  yrs.    8.  $  1000  ;  $626.  12.  2(J;  12;  10;  lOmUes. 

13.  126  cu.  yds.  14.  M,  390  ;  H,  130.  15.  39  ;  41 ;  32  ;  27. 

16.  3205  ;  2591 ;  1309.  17.  20  miles ;  4  miles ;  48  miles. 


156 


A   FIRST  BOOK  IN  ALGEBRA. 


1.  x  +  9. 

2.  a -{-p. 

3.  86. 

4.  x  +  y. 
6.  c  +  6. 


Exercise  10. 

6.  dx. 

7.  m  -\-  I  +  V  +  G  dols. 

8.  ic  +  y  +  ^  yrs. 

9.  bm. 
10.   d+1. 


11.  y  +  z  +  s  cts. 

12.  m  +  1. 

13.  yx. 

14.  X  +  40  +  a. 

15.  28 ;  46. 


1.  a  —  b  or  b  —  a. 

2.  6-10. 

3.  a  +  b  -c. 

4.  a  — 2,  a  — 1,  a,  a  +  1,  a  +  2. 

5.  a  —  6  dols. 

6.  c-8. 

7.  X  -  3,   X  -  6,  x-9. 

8.  c  —  6  dols. 

9.  x-5. 
10.  X,  ic  +  9,  or  x,  X-  9. 


Exercise  11. 

11.  X  -  75  dols. 

12.  w  +  X  dols. 

13.  c  -  /  cts. 

14.  b  —  e  dols. 

15.  I  +  4l  +  m  —  X  dols. 

16.  c  -  a  -  6. 

17.  429  ;  636  votes. 

18.  m-\-x  —  y  +  b  —  z  dols 

19.  80 -c  dols. 

20.  X,  60  -  X. 


1.  2x. 

2.  xyz. 

3.  100  X  cts. 

4.  a6c. 


Exercise  12. 

5.  a(^  cts. 

6.  mb  miles. 

7.  ax  hills. 

8.  x8. 


13.  jf  0  mx  dols. ,  or  6  mx  cts. 

14.  3 c  —  8  boys ;  4c  —  8  boys  and  girls. 
17.   5 6  fifths.  18.   m  -  X  +  2  a  dols. 


9.  a9. 

10.  d'^. 

11.  3w3  +  a2, 

12.  xs*"  -f  x"*. 

15.  9x  days. 

16.  3x  thirds. 
19.  12  a -39. 


1     ^ 
3c' 

2.  JLdols. 
100 

3.  -books. 


Exercise  13. 


4.   ^days. 

y 


6. 


b 
«  +  6 


7.  a  + 


8.    a  +  ^,  or^a. 

2         2 


9.   300  X. 


ANSWERS.                                     157 

10. 

186- 

Sxdols. 

11.   A,l;  B,l;  C,  ^  ;  all,  ^  +  -  + -• 
'  x'       y         z          X     y     z 

12. 

a^sq. 

ft. 

13.    100a  +  10  6  +  25ccts. 

14. 

y 

15.  -  chestnuts.           16.    12  ;  18  apples. 
m 

Exercise  14. 

10. 

11. 

12. 

21.                  14.  46.                      16.    -If 

11. 

7. 

13. 

78.                  15.    -74.                  17.    -4J. 

18. 

6. 

19. 

6  apples  ;  12  pears.                        20.   36  years. 
Exercise  15. 

1. 

24  a;. 

8. 

10 ax -4 6c.                 15.   5a +  46  + 5c. 

2. 

25  aft. 

9. 

-  16a2.                        16.   x  +  y-2. 

3. 

-18a«». 

10. 

8a«6  +  3a6-x6.          17.    -  32  -  a. 

4. 

-  42  a;. 

11. 

la.                               18.   2x3  +  4x2-2x+17, 

5. 

10  a2. 

12. 

-j\b.       '                  19.    a3+6»  +  c». 

6. 

-  10  a6c2. 

13. 

m  +  d+c-xcts.       20.   2a"*+l. 

7. 

Qah- 

■X2. 

14. 

a_a;-5  +  y  miles.  21.  2a262c. 

22. 

23x8 

-20 

x2  4-  27  X  +  6.                 26.   mb  -\-  c  men. 

23.  3fiy  +  12  x*y'-  -  16  xV  -  8  xy^.       27.   x  -  10  cows ;  «  +  19  horses. 

24.  6x  +  3y  +  2  -  a  -  3 6.  28.   22  girls  ;  30  boys. 

25.  a8  +  68  +  c3-3a6c. 


1.  2a». 

2.  12a26. 

3.  -9xy». 

10.  4x-y +  2«. 

11.  8x*  -  2x»  +  4x2  -  15x  +  14 

12.  20a262+16a26. 

13.  4x«-2. 

14.  2x*-x*»-x»*. 

15.  2a2'»- 18a'W-9x*». 

16.  ia^-la-h 

17.  -2x«y-3x»y«  +  5xy« 

18.  x-y  +  a. 


Exercise  16. 

4.  4x**y. 

5.  8x2 -3 ax. 

6.  5xy  +  7  6y. 
19.    -3a2 


7.  2a"». 

8.  9a2x. 

9.  _.«}„- 6  + 14c. 
20.    8x8  -  2 X. 

21.  27y8_3z8-6x«+4y22-ll0«x. 

22.  4a^-16x  +  64. 

23.  _4a2  +  662-86c  +  6a6. 

24.  2x*-3x'^  +  2x-4. 

25.  6a*  +  2a +  2. 

26.  -  1 1  a26  +  4  a62  -  12  a^b^  -  b\ 
y*.    27.   6 -a.  28.  x  -  3. 


29.  40  -  y  yrs. 


2JhiB. 


158  A   FIRST  BOOK  IN  ALGEBRA. 


Exercise  17. 

1. 

2x  +  a  +  b-{-G- 

-c?. 

5.   46 -4c. 

17.  5(x-y). 

2. 

a+  c. 

6.    -2y. 

18.    150-7(x  +  2/), 

3. 

2a^h-a^- 

-2  63_ 

-a62. 

7.    _66  +  4c. 

,     19.    x  +  8yrs. 

4. 

Sxy-x^- 

■3  2/2. 

8!    -b. 
Exercise  18. 

20.    3  (x  -  35)  dols. 

1. 

35  ex. 

5. 

18  acx^y^. 

9.    -f«2ca;V. 

2. 

—  51  acxy. 

6. 

30  a3&2c4. 

10.    -^%a365c4. 

3. 

21  ax^y^ 

7. 

-  X  V^8. 

11.    10«. 
ab 

4. 

10  a^^cK 

8. 

-  a4&5c2. 

12.    100  X. 

13. 

100  a  +  10  &  +  c. 

14. 

X  +  7  or  x  -  7. 

Exercise  19. 

1.  x^y"^  +  x^y^  +  xY-  5.    a552  _  _6^  ^^453  _  2  ^fSft*. 

2.  a*6  -  a3&-2  +  a^b^  '  6.   x^  +  ?/3. 

3.  -  2 a*b -\- 6 a^b^  -  2 ab^.  7.   x5-4x*  +  5x3-3x2+2x-l. 

4.  24  x42/2  +  108  x3y3  +  81  xy^.  8.   x^  +  x*  -  4  x^  +  x2  +  x. 
9.  x2?/2  —  2  x?/2n  +  yH'^  —  mhi^  +  2  xwi^w  -  x2m2. 

10.  x7  +  x6  +  2  x5  +  x2  +  X  +  2.  13.   x'  -  y''. 

11.  a«  4-  66.  14.  x8  -  8  x^a*  +  16  ««. 

12.  x5  -  3  x?/0  +  2/3  +  z^  15.    a6  +  2  a^^/^  _  9  ^^4^4  ^  ^6. 

16.  x^  +  x5  +  2  x*  -  11  x3  -  17  x2  -  34  X  -  12. 

17.  6x6-17x5- 12x4- 14x3  + x2  + 12x  + 4. 

18.  6{x  +  y);  4(x-y).  20.  1  +  i 

19.  II  of  the  field. 


a 


Exercise  20. 

1.  x2+9x+14.  6.   x2  +  4x-21.  11.  y^-2y-6S. 

2.  .  a;2  +  7  X  +  6.  7.   x2  -  X  -  42.  12.  x2  +  20  x  +  51. 

3.  3.2  _  7  X  +  12.  8.  x2  -  X  -  30.  13.  y^  -  13  ?/  -  30. 

4.  x2-7x  +  10.  9.    x2-13x  +  22.  14.  y^ -h  IS y -\- 32. 

5.  x2  +  3 X  -  10.  10.    x2  -  14x  +  13.  15.  a*  +  2  a2  -  35. 


ANSWERS. 

161 

16. 

a2-81. 

21. 

w»2  +  S  w  -  |. 

26. 

15-8x  +  x'^. 

17. 

m*-  18mH32. 

22. 

«^  +  ia-Vv. 

27. 

42  -  a;  -  x^. 

18. 

fe«  +  2  6»-120. 

23. 

x'-ix-^l 

28. 

33  +  8a;-x-^. 

19. 

x2-|x  +  J. 

24. 

y'+yy+ A- 

29. 

x2-9. 

20. 

y«+iy  +  T^,. 

26. 

21-  10x  +  a;2. 

30. 

y2  _  25. 

31.   21 

•• 

32.    12 
Exercise  21. 

cows. 

1. 

0^62.                 2. 

a^2/«. 

3.   a862.. 

4.    -«V. 

5. 

27  oV. 

10. 

121  c^''(P*z\ 

15. 

a»868ci«d8. 

6. 

49  a«6*c». 

11. 

\  x*a^m\ 

16. 

-  rB«V2^^mWn6. 

7. 

a*y««w. 

12. 

^  aW. 

17. 

5  a*62c8. 

8. 

w8n*d*. 

13. 

225ci2(Pa:4. 

18. 

r,  m^n*o(fi. 

9. 

-  125x9yi2;j3. 

14. 

-n^^y^^ifi. 

19. 

8  6  days. 

20. 

10 a  mills;     *    dols. 

Exercise  22. 

1. 

2. 

3. 

4. 

5. 
11. 
13. 
16. 
16. 
17. 
18. 
23. 
26. 
26. 
27. 
28. 
29. 


^»  +  32;2x  +  32;x2  +  x3. 
a«+4a^+6a2y2+4ay«+y*. 
x*-ix*a-\-6x'^a^-4xa*-\-a*. 
a«  -  3  a^m  +  3  am"^  -  w«. 
m2  +  2  am  +  a*. 
ar*y2  +  2x^z  +  ^2.        12.   rt854 
(fi-S  a^b»c  +  3  a266c2  -  ft^c*. 
a:»  +  3x2  +  3x  +  l. 
»n2-2ni  +  1. 
68_4ft6^.6ft4_4  52^.i. 

y»+3y«  +  3y»+  1. 


6.  x2-2xy +  y2. 

7.  X'^  +  3  X<y2  ^  3  a.2y4  ^  y6. . 

8.  w»«  -  2  m»y2  +  y*. 

9.  c8-4c«cP+6c<d«-4c2cr+(l». 
10.    y«  +  3  y«2r*  +  32/22?8  +  2I2. 

-  4  ofib^c  +  6  a*62c2  _  4  as^cs  +  c*. 
14.   x*y2  _  2  x^mn^  +  «i^n«. 

19.  a262_4a6  +  4. 

20.  x*y^-Qx^y-{-9, 

21.  l-4x  +  6x2_4a:8^.a:4. 


22.   I_3y2  +  3y*-y«. 
4x2  +  12xy2  +  9y4.         24.    27  a'^b^  ~  27  a'^b^^y  +  9  abx*y^  -  x*y>^. 
260  m*n^^  -  768  m»n»a25  4.  SQimhi'^a^b^  -  432  winVfc*  +  81  a^b*. 
1  x2  -  xy  +  y*.  30.   lOOx  -  a2  cts. 


1  -x^-\-l7^-ij7*.  31.   a(25- 

x8-12x«+54x*-108x«+81.      32.  3. 
20a2-d  horses.  33.   -228. 


x)  cts. 


160 


A    FIRST  BOOK  IN  ALGEBRA. 


1.  14a:- 7. 

2.  68  _  2  64  +  1. 

3.  10x''  +  7y2. 
10.  2x2-8x4-26. 
12.  x3-3x2+2?/-6. 
14.  x*-l. 

17.  watch,  $200;  chain,  $150. 

19.  8  ay. 


Exercise  23, 

4.  $160;  $80;  $6( 

5.  13;  21. 

6.  2  a3  +  4  a2  +  lo. 


7.  J  a2  _  4  «5  _|_  ^  ^,2. 

8.  48a766c7. 

9.  ifxV2;^2. 


11.    8  a663  _  36  a*62xy  +  54  a26x2|/2  _  27 xV- 
13.    x6-3xV +  3x2y4- 1/6. 

15.    x*  -  yK  16.    176^  lbs. ;  140^  lbs. 

18.   2x+4. 
20.    l  +  §6  +  i62-ia2. 


Exercise  24. 

1. 

bxy. 

6.    -llx^y. 

11.    -6x^yK 

16.    -6x2y223. 

2. 

13  a6. 

7.   4x^2. 

12.   -5a2c4. 

17.    2(x  +  ?/)202. 

3. 

3a2. 

8.   9a2c2. 

13.    |x3y. 

18.   5(a  -  6)2x. 

4. 

bx-^yK 

9.   2a26». 

14.   -Sa^m^. 

19.   ^xY^^- 

5. 

-17x. 

10.    -3x2y. 

15.  8m3x4. 

20.    -Aa263. 

21. 

x9?/6-9xV 

'  +  27x^?/*-27xV. 

22. 

A  miles. 
2a 

23.   ^ 

X 

days. 

24.    «^- 
c 

1.  3a62-76-f-15a3a;. 

2.  5  x2?/  +  3  y  -  9  x?/3. 

3.  8 x3«/5  -  4 x2y2  _2y 

4.  13a26-9a62  +  7  6. 


5.    -  f  a2x2  +  ^ 
11.   §a-^6-c. 

14.  ^minutes. 


:ax3. 


Exercise  25. 

6.  -|x2  +  2y2. 

7.  -4ifz<^  +  3x'^y^z*  -xy. 

8.  20  ac  -  31  a264c2. 


9. 


I  x^  +  .]  X*  —  4  X 


X2. 


10.    8  +  3/  ?/*  -  16  ?/3. 
12.    3x-2y-4.  13.   xi/ men. 

15.   1»«*  apples. 

X 


1.  x-7. 

2.  x-3. 

3.  X24-6, 


Exercise  26. 

4.  y2  _  6.  7.  x'^  +  xy  +  y2. 

5.  x2  -  5  X  -  3.  8.  a2  _  ^^  +  52. 

6.  a2-f2a-4.  9-  8aH12c?26  +  18a62+27i 


ANSWERS.  161 

10.  27a;^  +  9x»y  +  3xV  +  y^  16.  x*  -  a^  -  x^. 

11.  x»  +  ar^y  +  3  x^/S  +  4  |/«.  17.  a^  +  a''  +  a'^. 

12.  a«  +  4a26-3a62-2  6».  18.  a^o  -  a^  +  a^  -  a«. 

13.  x»  +  2 X*  -  3x  +  1.  19.  x'2  +  2  xi/  +  2 y«. 

14.  x«-3x-J  +  x-l.  20.  2-o2_6a6  +  962. 
16.  a^-a-1.  21.  ix^  +  xy-^y*. 

22.   ^x-^-ixy  +  |y2.  23.    J  (x  -  y)*  -  (x  -  y)*  -  H^^  -  J/). 

24.  2x2 -3y.  32.   2  +  f  a  +  Ja^  +  fa' 4.  etc. 

25.  4x«--4x-^-6x  +  0.  33.   3  -  |  ic  +  ^x^  - /yx^  +  etc. 

26.  2a^  +  a^b-2  aU'-  -  h\  34,  £  hrs. 

27.  a»  +  a262  +  6».  ^ 

28.  x*  +  xV  +  2/».  35.   ^±J^^±^dol8. 

29.  3a»  +  2  62.  «m+lp  ets 

30.  l  +  2x-2x2  +  2x3-etc.  ^n+p 

31.  1 -a -a--a^ -etc.  37.   y-llyrs. 

Exercise  27. 

1.  4n6«.  4.    -2a-65.  7.   2x2j/.  10.    §a«62. 

2.  3xy2.  5.   3a62.  8.   3xy.  11.    -fxV- 

3.  -'Ix^y.  6.   3xy».  9.    §«iV.  12.    - 1  a68. 

13.  x2(a-6).  15.   2a6»(x2-y)2.  17.   ?  a26«. 

14.  a»(x2+y2).  16.  4x2y  (ni' +  y)3.  18.    *x2y. 
19.    10  a'6»c*.                     20.  4y.                                   21.    \bxh/z\ 

22.   66.  23.    -^hrs. ;    -^1^  miles. ;    -«i^  miles. 

J:+y  a;  +  y  x+t/ 

24.  20  26.   2(m-6);  2m -6. 


Exercise  28. 

1.  2x-3y.  6.  x2-2x-l.  11.  x^-f  2x2 -f  x- 4. 

2.  x2  +  6xy».  7.  x2  +  3x  +  4.  12.  2x'-x2-3x  +  1. 

3.  4a6c2-7xy2«.  8.   2x2- x  + 2.  13.  90  -  x. 

4.  \x  ~  yH.  9.   3x2  +  X  -  1.  14.  lox  +  y. 
6.   06'  +  i  c*.                   10.  x«  +  x2  -  X  -f-  1. 


162  A    FIRST  BOOK    IN   ALGEBRA, 


Exercise  29. 

1. 

3x-y. 

6.    1  -  7  m. 

11. 

X2  -  X  -  1. 

2. 

5x2-  1. 

7.    4x2-1. 

12. 

1  a2  +  2  a  -  1 

3. 

36-^  + 4a. 

8.   3x3+1. 

13. 

4x  in. 

4. 

x2  -  2  2/2. 

9.   2a2_  a-f  1. 

14. 

27x3. 

5. 

1  +  Sz. 

10.   x3-x2  +  x. 

15. 

4?/ ft. 

16. 

a  -  b  miles  north  ;  a  +  &  miles. 

Exercise  30. 

1. 

45. 

5. 

8.4.                   9.    3.9. 

13.   3.28. 

2. 

97. 

6. 

.95.                 10.    73. 

14.    50.5. 

3. 

143. 

7. 

308.                 11.   62.3. 

15.    5.898-. 

4. 

951. 

8. 

.0028.             12.   83.9. 

16.    2.646-. 

17. 

.501  -. 

19.    74  men. 

21. 

92  trees. 

18. 

33  pieces. 

20.    104.9+ in. 
Exercise  31. 

1.  5(a2-5).  4.    15a2(l -15a2).      7.    a(a-b^). 

2.  16(l  +  4xy).  5.   x2(x-l).  8.   a(a  +  b). 

3.  2a(l-a).  6.    a'^(,S-ha^).  9.   2^3  (3^.^  +  2  a2). 

10.  7x(l-x2  +  2x3).  13.    (x  +  y)(3a  +  5  7n&-9d2x). 

11.  x(3x2-x  +  l).  14.    5(a-&)(l-3x?/-a26). 

12.  rt  (a2  -  ay  +  ?/2).  15.   ixy  (x"^ -Sxy  -  2y'^). 

16.  2ax?/5(3x2-2x?/  +  ?/2-a2/4).    18.    3a6(2a&-a262c- 3  62c  +  c2). 

17.  17x2?/(3x3-2x22/  + ?/3).  19.   3ax(x5  -  8  +  3x*- x3- 3x5). 

20.  27  a-562c3(a3  _  3  a2&  +  3  a62  -  ^3  _  ^3). 

21.  m  +  cZ  +  c-xcts.  22.    12  beads. 

Exercise  32. 

1.  (a  +  6)(x  +  |/).        4.    (x  +  5)(x-a).  7.    (x3  +  2)(2x  -  1). 

2.  (x  +  6)(x  +  a).        5.   (a-x)(x  +  b).  8.    (m-n)(x-a). 

3.  (a-6)(x2  +  ?/2).      6.    (x-4y){x  +  my).      9.    (x^  +  \)(x  +  l). 


ANSWERS.  163 

10.  («/•-+ l)(y- 1).  15.  (2a  +  36-c)(x-y). 

11.  (x«-x2+ l)(x+ 1).  16.  Sa(2x  +  y)(m-n). 

12.  (az  +  by  +  c)(a  +  b).  17.  250  4  y  A. ;  30  -  x  horses. 

13.  (a-b  -c)(x-y).  18.  12. 

14.  {3a-2b)(x  +  y).  19.  70. 

Exercise  33. 

1.  (x  +  y)(x-y).  3.   (ab^  +  cd)(a62  _  cd). 

2.  (?»  +  n) (m  -  n).  _  4.   (mp^  -  T^f-) {mp'^  +  Tfiy'^). 

5.  (a«6a:-J  -I-  m Vy*^)  Ca^?>x'-  -  ni-chj^'). 

6.  (ary^s^  +  cdm^)  (xy-z-  -  cdm'^).     9.  (2  a  -  3  x)  (2  a  +  3  x). 

7.  (x*y  +  aV-)(a:-y-aV)-  10.  (4  wi  -  3n)(4m  +  3n). 

8.  (9«c8  -  a:«2*) (^'c3  +  x^z*).  11.  (9 ary^  +  6  bd) (9 xy2  _  6  6(i). 

12.  (27  mkx^  +  100  j/'^)  (27  m-cx*  -  100  ?/2). 

13.  (llm  +  8x)(llwi-8a:).  14.   (x'i  ^  y^)(x  +  y){x- y). 

15.  (m*  +  a*)(m2  +  a2)(„i  +  a)(„i  _  «). 

16.  (a*b*  +  l)(a262  _,.  i)(a6  +  i)(a6  -  1). 

17.  (X8  +  68)  (a;.  +  54)  (x2  4.  52)  (jc  +  6)(x  -  6). 

18.  (4a2  +  l)(2a  +  l)(2a-l).  22.  5a(x  +  y)(l  +  a)(l  -  a). 

19.  rt(«  +  r)(a -x).  23.  (m-h  y  -  a)(m  -  y). 

20.  5  62(6  +  a)(6-a).  24.  (a- x- l)(a  +  x). 

21.  (a- 6)(x  +  y)(ic-2/).  25.  x-3?/.        26.   Sx+llyrs. 

Exercise  34. 

1.  (x  +  y) (X-  -  xy  +  y2).  3.   (a  +  6c)  (a^  _  a6c  +  b^c^). 

2.  (c  +  d)(c2-cd  +  (P).  4.   (ax  +  y)(a2x2-axi/-f  y2). 
6.   (2  a6c«  +  »/»2)  (4  a^b^c*  -  2  a6c%2  +  m«) . 

6.  (x2y8  +  6  a)  (x^y''  -  6  ax^y*  +  36  a^). 

7.  (a2  -f  6*0(0*  -  o'fc'^  +  6*).  10.   (3  +  a62)(9  -  3a6^  +  a^b*). 

8.  (4x2+y2)(i6a;»-'4xV+y»)-    H-    (y  +  l)(y=^  -  y  +  !)• 

9.  (x  +  2)(x2-2x  +  4).  12.   (l  +  6c«)(l-6c8  +  6V). 

13.   (ia26  +  c»)(ia«62- Ja26c8  +  c9).    14,  (Jx  +  l)(ix2  -  ^x  +  1). 


164  A    FIRST  BOOK  IN  ALGEBRA. 

15.  (m  +  71  +  2){(w  +  w)2  -  2(m  +  n)  +  4)}. 

16.  (l  +  x-y){l-(x-y)  +  ix-yy}. 

17.  2  a2(a:?/2  +  a^b^){x^y*  -  a^b^xy'-  +  a*66). 

18.  (m  -  n){x  +  y).  19.   (x  -  2/)(a  +  6)(a2  -  a6  +  6-2) . 

20.  (a  +  &)(a2-a&  +  &2)(a-6)(a-^  +  a6  +  62). 

21.  (27x3  +  4y3)(27:K3_4y3).  22.    (l  +  x*)(l  +  x2)(l  +  x)(l -:^:). 
23.    3(d-26);   3c?-104dols.           2.4.    10  y.  25.    a:  -  8. 

Exercise  35. 

1.  {x- a)ix^ -V  ax  + a"^).  4.    (ww  -  c)(7w-w2  +  mwc  +  c^). 

2.  (c  -  6) (c2  +  c6  +  5-2).  5.    (3  wi_2  a-) (9  iii'+Qmx+^x:^). 

3.  (a  -  a;?/) (a2  +  axy  +  a:2^,2).  e.    8(a;  -  2  //)  (^2  +  2  xy  -^iy'^). 

7.  (4  ax26  -  5  m'^cy^)  (16  a2a;i62  +  20  aban^xh/  +  25  m^'chj^). 

8.  27(6c2|/  -  2  a3|,i2^2) (52^4^,2  ^2  a^bchn^x^y  +  4  « '«i+x*). 

9.  (2  a;3y  -  5)  (4  xV  +  10  a:^!/  +  25) . 

10.  (3-4  mx^y)  (9  +  12  mx^y  +  16  m^x'y^). 

11.  (a- 62)  (^2  + 0,52  ^_  54).  13.   (^x-l)(x^  +  x-\-l). 

12.  (?/i2  -  a:)(?/i*  H- ?>i2a;  +  iK"^).  14.    {I  —  y){l  +  y  +  y'^). 

15.  (ia;?/2- 63>)(ijc3yt  +  ^a;y253  +  66). 

16.  (2  -  i  w2?0 (4  +  ^  ?)i2^  +  1'^  w*n2). 

17.  {l-(a  +  6)}|H-(a  +  6)  +  (a  +  6)2}. 

18.  {xy  -{X-  y^)}{xY'  +  xy{x  -  y^)  +  {x-  ?/)2}. 

19.  x(a;2-?/)(a:* +  0:2^  +  2/2).  20.    {a  +  m) {b  -  c) (b- -i- be  +  c"-) . 

21.  (a:  +  y)(x2-  xy  +  y^){x-y)(x^-hxy-\-y'^). 

22.  (x5  +  x2-  l)(x  -  1). 

23.  {a  -  b)(a-j-b-\- a''- +  ab  +  b^).' 25.   —  hrs. 

xy 

24.  (x  +  ?/)(wx2-mx«/+?w?/2-l).      26.    1002+10x  +  ?/. 

Exercise  36, 

1.  (a  +  x)2.                    5.   (x-3c)2.  9.  (x  -  5)2. 

2.  (c-d)2.                    6.    (4x-2/)2.  To.  (2?/ -3)2. 

3.  {2a  +  yy\                 7.    (a  +  1)-.  H.  (3x  +  4)2. 

4.  (a +  2 6)2.                8.   (3-x)2.  12.  (3a-l)2(3a  +  l)2. 


ANSWERS.  '  165 

13.  (3c  +  lld)2.  17.  (4ary2_3a8)2.         21.   (ia  +  26)2. 

14.  (2y-9)2.  18.  (5  6  +  3c-iy)2.          22.   (2  x»y -«*)'. 

15.  (a;2  -  3y)«.  19.  (7  a  +  2x-'y)«.          23.   x3(a:3  _  3^4)2. 

16.  (3-2  a;2)8.  20.  (^  x2  -  y:)2.             24.   2  ?/(3  a  +  2  y2)2. 
25.  3aa:»(aa;-5  6)-.                         26.   (x -\- y  -  a^y\ 

27.  {(m2  -  n2)  -  (W^  +  w^)^  or  (-2  n^)'^. 

28.  (a  +  6)(a  +  6  +  6).  .     30.  {x  -  y^^x'^ -{■  xy  +  y^^. 

29.  (x-y)(x-y-  3).  31.   y2. 

32.  9y2.  36.  4y*.  40.  i6a2  or  28  a^. 

33.  ±2  erf.  37.  2b y-.  41.  ay  or  353 «y. 

34.  1.  38.  ±00 ab^c.  42.  4a*-'6-2  or  lOa'b'-. 

35.  9.  39.  ±  70  a^ftScd.  43.  x^  or  -3  a:-. 

44.    a  or   -  3  a.  45.   6  -  a  ;  0(&  -  a)  ;  6  +  a  ;  3(6  +  a)  miles. 

46.   i»5  of  the  cistern.  47.   ^-^ ;  ^-=-^  +  9  yrs. 

o  O 

Exercise  37. 

1.  (a  +  2)(a+l).  3.   (x-3)(x-2). 

2.  (a:  +  6)(x  +  3).  4.    (a-5)(a-2). 

6.  (y-8)(y-2).  9.  (y+l3)(y-5).     13.   (a-ll)(a-13). 

6.  (c-3)(c  +  2).  10.  (a-ll)(a  +  7).     14.    (a8  -  13)(a8  + 9). 

7.  (x  +  5)(x- 1).  11.  (a:-9)(a;  +  7).       15.   (a;^  +  11)(«8- 7). 

8.  (x  +  6)(a:-l).  12.  (a  +  15)(a-6).     16.   (6  +  x)(5  +  a;). 

17.  (7  +  a)(3  +  a).  18.   (7  -  x)(5  -  x),  or  (x  -  7)(x  -  5). 

19.  (9-x)(4-x).  24.  (x2-6|/2)(x+3y)(x-3j/). 

20.  (c  +  3rf)(c-d).  25.  x8(x-12)(x-ll). 

21.  (a  +  5x)(a  +  3x).  26.  a*(a  -  7)(a  -  6). 

22.  (j:-5y)(x  +  4y).  27.  3a(x+3)(z-8)(x+2)(x-2). 

23.  (x2  +  4y)(x2-3y).  28.  3a(a-2  6)2. 

29.  (c  -a)ic  +  rt)(cd  -  l^c^d^  +  cd  +  1). 

30.  («  +  6)(x-3)(x-2). 

31.  -  +  -•  32.   ^days. 
a     b  2 


166  A    FIRST  BOOK  IN  ALGEBRA. 

Exercise  38. 

1.  9a'^  -9a  -4.  3.   x^  -  yK 

2.  Sy^-^2y^-2y-l.  4.    a^  -  5a2  +  26a  -  2. 

5.  x^y^  -  12  xY^z^a^  +  48  x'hjz^a^  -  64  z^a^-\ 

6.  (x-10)(x-l).  9.    x^(Sx^-n)^. 

7.  (ia  +  b  +  c  +  d)(a  +  b-c-d).    10.    (2c2  -  #)(4c4  +  2c2d3  +  (?-). 

8.  (2a:2-3)(3x  +  2).  11.    (1  +  4a;)(l  -  4a;  +  16:»:2). 

12.  (a  +  l)(a-^  _  a  +  l)(a  -  l)(a-2  +  a  +  1). 

13.  (x  +  2){x-\-l){x  -1).     U.  (x-y){x^  +  x^y  +  x'2y^  +  xy^  +  y*). 

15.  (9  +  a:)(3  4-x).  17.    -  3m2  +  8m?i  +  3n2. 

16.  ?w2  +  2m-4.  18.   52-25^-52x2. 

19.    2x3.  20.    18;  19;  20.  21.    60.  22.    16. 

Exercise  39. 

1-   3a62.  4.   Sa^b(a-b).       7.   x  -  5.  10.  y(y-l). 

2.  5a:y.  5.    m-n.  8.   x  +  3.  11.   x  -  y. 

3.  2xy^{x-\-y).      6.   9x*-4.  9.   a  +  2.  12.   Sa(c:^-a^). 

13.   a:2(2a:3-i/2).  14.    ^^±-^  cts.  ^^-    rfo^^dols. 

«  +  ^  16.    5;  11. 

Exercise  40. 

1.  (a2-4)(a  +  5).  11.  (1  _  a^)2. 

2.  (0:3+ l)(x  +  l).  ,     12.  6ax2(a2_a;2). 

3.  (x2-9)(x-5).          ■  13.  (m2-w2)(a2_i0a  +  21). 

4.  (x6-l)(a;2_3).  14.  6te(l-63)(l  +  5). 

5.  2(a:2-l)(a;-3)(x  +  2).  15.  bab(a+x){a-xy^ia-x-l). 

6.  (a+l)3(m-2).  16.  c +  ?/ degrees. 

7.  Ca2-4)(a2_.  25)(a2_9).  17.  6  -  ?/,  or  ?/ -  6  degrees. 

8.  (X- 1)3(2/ +  3).  18.  ?. 

9.  (x2- I)(x2-9)(x2-16).        19.    x3+12. 

10.    2x3(x3+ l)(x  -  1).  20.    $306;  ."^  1836. 


ANSWERS.  167 


Bzerclse  42. 

1     '1^.  6        ^  9    ^^(^  ~  ^^^      13     «(^  -  ^) 

*   8  2  *    a2-  l'  a«       '         '      x  +  o    " 


2  a;*  g        a;2 


10.  y(«  +  ^y).  14.  «i^-^. 

3  ^  — ^-              7     m  +  n           ^^     a  -  b  ^g     !_+_?. 

'   x-{-5                '   a  +  2b              '    a  +  b  '   2  +  x 

4  31±6.         •    8      ^  +  ^            12    ^'  +  y'.  16.   l^L^. 
'a;-5                'ic,^2y                  x^-y^  i_6 

17.    j  6,  or  ^  cts.  18.  ?  yrs. 

Exercise  43. 

1.  ft_c+--           3.  a;  +  y  +  -^-              5.  2a6c-36  +  4.- 

X                              X  —  y  a'^b 

2.  m  +  a  +  -.           4.  a^-ab-i- b^- -^-     6.  3 x^  +  5 x^  - -^. 

n                                         a  +  b  xy^ 

7.    X  +  3  4- -^-^^ti_.  11.    4a2  +  2a  +  H-       ^ 


x-i  -  X  -  1  2  a  -  1 

8.    2a-l+    ,^^~^    •  12.   9x2-3x4-1-      ^ 


a2  +  a-2  3x  +  l 

9.   x2  +  xy  +  y2  +  -2yi..  13.   3x+2-    .^  ^"^      ■ 

X  —  y  ^                               x^  +  2x— 1 

10.    a3_«5  +  62-l^.  '       14.    2a -3+    /^«  +  ^    . 

a  +  6  a'^  +  3rt  +  2 

15.   2o2  +  4a-2.  16.   3x2-2x+l. 

17  ^^-y^-^y        19  2d^ 21.  ^-^^-'^^' 

X  -  y  c2  +  cd  +  cP                        3  X  -  1 

j8^    g^  -  2  a6  -  6^        2Q  x«  +  y^                     22.    2  g^^  +  4  g  -  1 

a  +  6  x  —  y                                    a4-3 
23    2a*-6a«4-3a^4-2a-6      g^        a*  +  2a'^  +  2g4-3 

2  g^  -  1  '                        g2  _  a  +  3 
6x«  +  3x«- 7x2-1 


24. 
25. 


3x2+1 
4x-x2 


X2-X+2  2x 


28. 

x3-2x 

+  1 

X2  +  X 

- 1 

42. 

?^    Ct 

26     -     ^'^•^^^  43.   g  -  3,  g  -  2,  a  -  1,  g. 

g2-2g  +  3*  44.    2rn  +  2.  45.    4a-l. 


168  A    FIRST  BOOK  IN  ALGEBRA. 


Exercise  44. 

1. 

12* 

3. 

43  X 

60 

5. 

4x 
15* 

2. 

X  ^ 
15* 

4. 

4  am  +  3  bx 

2  bm 

6. 

m'^  —  2  m2  n2  ■ 
m2;i2 

+  ?i^ 

7. 

3  X  —  mx  +  5 171-71 

g     115-x. 

9    0 

o 
10 

rt^> 

Smn 

9x 

a2 

-62 

n 

&-2 

13. 

4^2 

15. 

2  ax 

x3  ~  8  a' 

{a -2  b)  (a-  b) 

X  (X  -  2  ?/) 

^^ 

2 

14. 

2a:2-4x  +  29 

(x+2)(x-3) 

16. 

2x9 

X5  _  if 

(x  +  5)(x  +  3) 

17. 

?7l2  +  11 

(77Z  -  1)  (m  +  2)  (i 

20. 

m  +  3)                   (a  - 

2 

2)(a-3)(a-4) 

18. 

0. 

21.   2. 

19. 

2(9xi+  1). 
9x^-  1 

22.        2     . 

a  +  3 

23. 

x  +  25 
x^-x-  20 

26. 

x2  +  xs  -  ?/s 
(x+s)(?/+0) 

29. 

a 

2 

24. 
25. 

0. 

2 
a 

27. 
28. 

0. 

X 

Exercise  45. 

30. 
31. 

1  X 

3* 
4x-21. 

1. 

2 

x  +  2 

7. 

b 

13. 

1  +  a  -  a* 

1  +  a 

5(2/4- 9  &2) 

2. 

3. 

4. 

5. 

a 
a'^  -  9 

1 

2a +1 

32/ 

X2  +  X?/  +  2/2 

8. 

9. 

10. 
11. 

1 
(l_a:)(3-x) 

6 
(&-c)(a-6) 

0. 

1. 

14. 
15. 
16. 
17. 

1 
(x-2)(x- 
43  a 

b 
9x 

4  ' 
mx 

5  ' 

-3) 

6. 

1 

12. 

2a2 
a2-l 

18. 

a 

3(a2  -  1) 

1. 

ox 

2. 

3  a-bc 

3. 

a* 
2c^* 

4. 

4a/>c2 

ANSWERS.  169 

Exercise  46. 
6.  ?^».  9.   3(x  +  y)2.        13.  a;.      15.    10. 

g    3x(x-y).     iQ    «--^«-21.    14.   ,,      16.  5x. 
2  mn  a  +  2 

7-  1^^^^-  11-  x^-2a:y+y^.    17.  24  a. 

Sx-y 

12.  a2+2a6  +  6^. 


Sx  rK^  +  ary  +  y'^ 


1. 

a 
b 

2. 

3m2n 

4x^y 

3. 

3fP 
6a; 

4. 

2x^v 

Exercise  47. 

6.  ^.  11.  a:  +  l  +  -L.    16.  -1^. 

^     a-  b^  12  X--  5x  +  6    J-    29 

6  ax:-yz  '        x  +  4  '35' 

8.  ^(^  +  ^).       13.        ^     ♦  18.  ^. 

a-y  12rt-'x3 

9.  ^ ,.  14.  2rtc  +  ^«'^' 


'> 


3m"n  2m(7n-n)2  2x^y 

10.     ^^"^    .  15.  ^y(^-y). 


3x(c  +  d)  a;-«-7a;  +  12  8^22 


Exercise  48. 

1.  «''-. 

8ca-?/ 

3.  2. 

6. 

cd 
ab 

2.2. 
3 

4.  3. 

6. 

2]/. 
an 

^    (a;2  +  4a;+16)C3a:  + 
x 

3. 

14. 

10.  m2  +  3m  +  9. 

11.  a. 

12.  (^-^)'. 
2x(x  +  3) 

15. 
16. 

26.     25 

2a 
3  c* 

262 ;  224. 

13.  ^«  +  ^)^ 

3a(a  +  2) 

• 

17. 

7  ^+i. 

'  x  +  1* 

8.   a  +  6. 


dols. 


170  A    FIRST  BOOK  IN  ALGEBRA. 

Exercise  49. 

^    6  a^bc  3  I6a^z^                         g     x^  -  25 

Txif'  '  9c%2' 

2_  21aW.  4_  4  7/iV                          6_ 

16  mnz^  9  a'^a;^ 

7.  ^.  11-    1-                     14.   i^ 


8. 


^  -  -3  2  n 

X  —  5 


q(a-  7)         12. 
'  a  +  6    ' 


a:  +  5  15. 


x^  -\-2x 

(X- 

-5)(x- 

6) 

(X- 

-3)(x- 

3) 

17. 

3. 

18. 

4. 

19. 

20. 

4  c' 

21  xAy^ 

^-   1-  13.   «.  16.  J-.  

10.    1.  &  .  4x  ""•   20«53 


Exercise  50. 

1    i^.  2    -^.  3  32x5j/5         ^    a;^(a  -  b)^ 

a^'b^'                  '   64?/3*  •       243ai'^mia'       '       81  a^d* 
g        125x8^3(^^^2)8  ^     _  27#?^i--^(2a  +  35)9 
—  yy  '  64  x^y\m  —  ny 


^    81  a20a;iy8^i2  ^^     6mn^^  ^3        3x2(a-&)3 

16  6+ci*^(?io  '  •   Ua'^i)S  '  4^254 

g    256x28^8^12  ^^    4x?/2  j^    2x  +  3y 

81a4624^12'  •     3(^^3*  •         4a;2^4 

9     2a&2  ^^         2xy2  ^g    2x(x  +  y)2 

3m2?i3'  ■       3m2w3  '  5y^ 

16         3a(a-&)2  ^^  x*        81  yj 

2  63  •  256      625  m* 

17.  ^-^.  23.    -+1. 

18.  ^'-^.  24.    ^'-1. 

62       d^  6-3 

19     6  r  -I-  3  abx^y  _  a'^x  05     ^  _  3a2x      9  a* 

2  ?/!«         m  2/2        5y        4  52 

2Q^    2x^_3x|/^_x^,  26.    it+^^y^  81. 


3?/0  2  »i2^2  c*  c2d         4(?2 

a_8 
68 


21.    ^'-^.  27.    «'-^-10 


ANSWERS.  171 


+  i^-f.  31.    f2i  +  ^W!L+»Wl-5y 

6*  \m^     x'^J  \m     xj  \m     xj 


y»       fey!*        6'^ 


34. 
35 


Exercise  51. 
5.    «±^.  9.    «±i.  13.    «'  + 


aft  a  —  1  2  a 

6.  ?n(m2-m  +  l).  10.   a^.  14.    1. 

7.  £ilLi^.  11.   ?^.  15.   I 
ax  —  by                        X  —  o  a 


12.    a.  16.    1. 


8.  -i 

c 

17.  6y  lbs.  19.    lOOy-x^cts. 

18.  12y  in.;   ^  yds.  20.    -. 

3  m 

Exercise  62. 
1.   33C*-2«»-a;  +  6.  2.   45;  65.  3.   x  +  1. 

4.    (x*-7y)2, 

(a  -  ft)  (a^  +  a6  +  ft«)  (a«  +  a«ft'  +  6«) ,  5.   2. 

3(a-9)(a  +  8). 

U-ft-^     OyVUft      3yJUft      3yj  ^^ 

8.    (ai-16)(a«-9)(a2-4).  9.    1?^.  10. ^• 

a  X  — 4 


172 


A   FIRST  BOOK  IN  ALGEBRA. 


11.    0. 
15.   1. 


12.   ^. 


2y 
17.   4:2  xy. 


13.    ^. 


18. 


2x^ 


14.    1. 


--^^x-l. 


Exercise  53. 


1.  z  =  2. 

2.  x  =  -3. 

3.  x=19. 
4:.  x  =  -  13. 

5.  x  =  2h. 

21.  x  =  -f. 

22.  x  =  l. 

27.  a;  =  51. 

28.  x=-  15. 

29.  a:  =  48. 

30.  x  =  ^. 

43.  —  hrs. 
15 

44.  ^'*  dols. 
100 

45.  16;  41. 

52.  x  =  0. 

53.  x  =  2. 

54.  aj  =  a  +  &• 

55.  x  =  f. 

56.  X  =  i:. 


6.  a;  =  5}. 

7.  :^  =  4. 

8.  ;b  =  6l. 

9.  x  =  l. 
10.   :»  =  3. 

23.  17 

24.  .$15. 

31.  x  =  -5. 

32.  x  =  -l, 

33.  a:  =  L 


11.  x==2. 

12.  x=l. 

13.  x  =  10f. 

14.  x  =  2. 

15.  x  =  l. 

22  ;  Q6.  25. 


26. 


34. 


46. 


35.  x  = 

36.  a;  = 

37.  x  = 

38.  x  = 


i^  dols. 

100 


47.    x  = 


48.    x 

57.  a:  =  l. 

58.  x=T-V 

59.  x  =  2. 

60.  a;  =  2. 

61.  x  =  4. 


a  —  h  +  2c 

—  1 

—  2' 

62.  a:  = 

63.  .x  = 

64.  x  = 

65.  ic  = 

66.  X  = 


49. 
50. 

51. 


1. 

tV- 
4. 

-7. 


Exercise  54. 

1.  209  boys;  627  girls.      6.    17  ;  22  ;  88. 

2.  184  ;  46  ;  23.  7.   9. 

3.  16  ;  28  yrs.  8.    9  fives  ;  18  twos. 


16.  x  =  3. 

17.  :c  =  4. 

18.  x  =  :^. 

19.  :«  =  -8. 

20.  x  =  l. 
28  X  fourths. 

—  days. 

z 

39.  x^2. 

40.  x=\\. 

41.  a:  =  6. 

42.  a:  =  8. 
x  =  3. 

2 

«  +  c 

67.  26  ;  27  ;  28. 

68.  6?/ years. 

69.  —men. 
a 

70.  12  a; +  3. 


11.  12  M.  ;  30  J. 

12.  80. 

13.  24. 


4.  36;  122. 

5.  S.,  5;  H.,  11  qts. 


9.    18  ;  15  ;  25  tons.       14.    48.- 
10.    15. 


ANSWERS.  173 

15.  t.,  5  lbs. ;  m.,  8  lbs.  ;  s.,  14  lbs.  16.   35  ;  10  yrs. 

17.  $2070,  S 920.  18.    17,042;  14,981;  15,496. 

19.  R.,  213  ;  J.,  426.        22.    45  ;  75.  25.   J.,  12  yrs. ;  S„  28  yrs. 

20.  63  ;  64  ;  65.  23.  24  ;  48  yrs.     26.   20  yrs. 

21.  16  ;  28.  24.   2  ;  12  yrs.      27.    A.,  6  yrs. ;  G.,  18  yrs. 

28.  Ed.,  40  yrs.  ;  Es.,  30  yrs.         30.  3  ;  12  yrs. 

29.  6  yre.  31.   2x^  -  Qx^y  +  IS  zy^  -  27  yK 

33.    — ^.  34.   x  =  7.  35.   $32.  36.   $13. 

x-if 

37.  A,  $47;  B,  $28;  C,  $61.        39.    u.,  $18  ;  cl.,  $25 ;  h.,  $9. 

38.  80  cts.  40.    J.,  $  1.80  ;  H.,  $2.42 ;  A.,  $1.25. 

41.  2i  day.s.  52.  x=  IJ.  58.  (1)  49^^  m. 

42.  (>^j  days.  53.  1.  (2)  l^f?  m. 

43.  22ihrs.  54.  ic«  -  13x*  +  36.  (3)  32^8^  m. 

44.  1 J  hi..  55.  (3a  4- 26)-  '^'    ^^  m.,  32^  m. 

45.  4|hrs.  (3  +  x)(4  +  .);  f^^^^ 

46.  2Hhrs.  (a- 26)(c- 3d).  61.   30/^  m. 

l\  (.«  n  ;  62    5^m.pastl2M. 

4/.  /?  nrs.  ^Q 

48.  171;  7^,  days.  '7^^^  ^'   ""^^^ '^' 

49.  um..  f^Zt  ^'  '''"• 

'^^  C-^)  ^^^  °^-  65.    24  hrs. ;  152  m. 

^^-  ;r+^  ^*y®-         57.  (I)  27,5^  m.  66.   35  miles. 

c-d    *^^'  (3)  21/y  m. ;  64^  in.  70^  miles. 

68.  24  miles.  69.  6  p.m.  ;  30  miles  from  B  ;  25 J  miles  from  A. 

70.  16  sec.  72.    12  ft.  by  24  ft.         74.    16  sq.  ft. 

71.  11  ft.  by  15  ft.         73.    14  ft.  by  21  ft.         75.   48  ft.  by  72  ft. 
76    f^  +  J?fiV?_J^V  77.   a:»-2a:2  +  4a;-3. 

[2     3m*)\2     3m«y*  78.   7y«  +  ISjc^y  -  x». 

(a:2_3y)(ac*  +  3x«y  +  9y2);  79.   a:(x  +  1). 

(a9  +  68)(a»  +  6*)etc.;  80.   60  ft. 

2c(c2  -  1)2.  81.  A,  7  days  ;  B,  14  days  ;  C,  28  days. 

82.  9 ;  48.  85.  24 ;  25.  88.    300  leaps. 

83.  18  ;  74.  86.   A,  67  yrs. ;  B,  33  yrs.      89.   With  144  leaps. 

84.  $140.  87.   38.  90.  After  560  leaps. 


174  A   FIRST  BOOK  IN  ALGEBRA. 

Exercise  55. 

1.  x  =  3,    ij  =  l.  1.  x-b,   y--2.  13.  x  =  l2,   ij  =  16. 

2.'  X  =  4,    ?/  =  2.  8.  x  =  2,  y=-l.  14^.  x  =  6,   y  =  18. 

3.  X  =  7,    ?/  =  6.  9.  X  =  -  21,    ?/  =  -  J.  15.  X  =  35,   y  =-  49. 

4.  X  =  5^,   y^  4.  10.  X  =  2^T,    ?/  =  -  1^  16.  x  =  21,   y  =  25. 

5.  x  =  L^,  |/  =  3.  11.  x=*,   y  =  lh  17.  x  =:  3,  ?/ =  2. 

6.  x  =  2,   y  =  S.  12.  x=15,   i/ =  -  17.  18.  x  =  ^,   y  =  4. 

19.  X  =  12,   y=i  15.  28.   27  ;  63. 

20.  x  =  ^i^,    y  =  ^!—^.  29.    13;  37. 

2  2 

21.  X  =  39,  y=-  50.  30.  J.,  13  yrs. ;  H.,  19  yrs. 

22.  X  =  -  2,   ?/  =  -  lif  31.   A,  36  cows  ;  B,  24  cows. 

23.  ^V  24.  {I.  25.  j\.  32.  tea,  54^ ;  coffee,  32^. 
26.  ii  27.  24  ;  62.  33.  corn,  61f  ;  oats,  37^. 
34.   12  lbs.  of  87^  kind  ;  26  lbs.  of  29^  kind. 


Exercise  56. 

1.  x=±3.      4.    x=±2.      7.   x  =  ±7.     10.  x  =  ±2.     13.   12  yrs. 

2.  x  =  ±2.       5.   x  =  ±^.      8.   x=±2.    11.  x  =  ±2.     14.  48 ;  64. 

3.  x  =  ±5.      6.  x  =  ±l.      9.  x  =  ±l.     12.  x  =  ±3.    15.    ii. 

Exercise  57. 

1.  X  =  3  or  -  6.  9.  X  =  1  or  -  a.  17.  x  =  5  or  -  2. 

2.  X  =  2  or  -  7.  10.  X  =  -  or  -  -.  18.  x  =  0  or  5. 

c  a 

3.  X  =  9  or  -  8.  11.   X  =  11  or  -  3.  19.   x  =  0  or  -  3. 

4.  X  =  7  or  3.  12.   x  =  G  or  1.  20.   x  =  3  or  3. 

5.  X  =  8  or  15.  13.   x  ==  5  or  2.  21.   54^-^  m. 

6.x  =11  or -17.        14.   x  =  6  or  4.  22.  J.,  $18  ;  L.,  |6. 

7.  X  =  &  or  6.  15.   X  =  6  or  16.  23.   3  in. 

8.  X  =  a  or  3  «.  16.  x  =  2  or  —  3.  24.   7f  ^  days. 


ANSWERS.  175 

Exercise  58. 
1.  3.  4.   80  hrs.    6.   a^  +  ab  -  b\  8.  60-a:2-28a:. 

3.  x  =  3.     5.  x^h     7.   l-»/  +  3y2  +  2/.     9.  a:=±C.        10.1. 

11.  ll(a-o)-17a:«//-3(a-c)8.  12.  x  =  i^. 

5 

13.  (a;3  +  3)(a:_i)(a;24.x+l);    (.a -\- b)ix  -  y){x  +  y)  ; 
(3a;  -t-  y  4-  z)  {9x2  -  3x(|/  +  z)  +  (y-\-  z^}. 

14.  W-       16-  a  +  ft.       16.  X  =  5.       17.  J.,  10  ;  S.,  8.       18.  ^—-^. 

X 

19.  a^_36«  +  3c2.  21.  rt+3x  +  4y  +  26.        23.   1^  days. 

20.  ^.  22.  a'^-ia-^  1.  24.  -A«L. 

25.  (r  -  13)  (X  +  4)  ;    (1  +  a^)(l  +  a*)(l  +  a-)  etc. ; 
(a*2  +  62  4.a-2-62)2or(2rt2)2. 

26.  x  =  36.  29.  3x*-2x'»  +  x-5.       33.   25;  26;  27. 

27.  00;  40yrs.  30.   2.3o ;  3.2.  34.   a((i-h2\ 

28.  x  =  3,   y  =  12.      31.  S^^.       32.  x  =  0.      35.  ^i±J^^        36.   f. 

3-        27  a%\m -\- n)\      5x-^(a  +  6)8 

64x-V(«-&)*'  4y»« 

38.3x2-2x4-1.  40.  x  =  -f,  2/  =  -?. 

39.   i.  42.   ix2y2  +  xy. 

41.  (x2  4-  3)(x2  +  2)  ;  (X  -  7)2  ;  (X  +  y  +  z)(x  -  y  -  z). 
43.  31xVm.;  44i^rin-  44.   1.  45.  x  =  b. 

47.  4idays.  48.  x  =  18,   y  =  6. 

49.  1  +  6x4- 12x2+ 8x»;  16x9-96xV6H216x*a*6'-216x2a«69+81a86". 

50.  «i±4?.  51.  y  -  X.  52.  0. 

53.  400  sq.  ft.  64.  2(a2  _  l)(a  -  3) (a  ~  2). 

55. -(i-f>   .  6e.  a;  =  4  or  4. 

2(2a  +  6) 

57.  abia  +  2  cm*)  (a«  -  2  aom2  +  c2m<) ; 

c(2cx  +  y)2 ;  (X  +  l)(x2  -  X  +  l)(x  -  l)(x2  +  x  +  1). 

58.  l-X-T!»jX2  +  |x»+iX»-^X«. 


176  A   FIRST  BOOK  IN  ALGEBRA. 


59.  x2  +  X  +  1.  61.  a'-  +  h  a^. 

60.  x^-^6xy  +  y^+    9xy^-6y^   ^  ^2.  x  =  ^^. 

x^  +  2y^-Sxy  a  +  b 

63.  (X  4-  5)  (a;  +  5)  (x^  +  3)  ;  (4  +  x')  (2  +  x^)  (2  -  x^)  ■  (2  a)  (2  6). 

64.  3|days.  65.  x  =  5.  66.  ^' _  ^o^c     3ac^  _  c^  .  c^ 

h^      b'-d        bcV       # '  #  ^  ^• 
67.   24   or   -  3.  68.   a;  =  15,   ?/  =  14. 

69.   m  -  my  +  my'^  -  mij^  +  etc.       70.  —- 

o 

71.  4  ft.  72.    10  a-  +  19  ax  +  19  ay  +  9  x'^  +  18  xy  +  9  y^. 

73.   |x2-x+i.  74.   5x*  +  4x3  +  3x-  + 2x+l. 

75.  (a  +  2)(a-2)(a-f  5);  (x -\- y) (x^  -  xy -^ 2/^) {x - y) (^x^ -\- xy  4- y"^) ; 
2x(x2-3?/2)(x  +  ?/)(x-?/). 

76.  8hrs.  77.  x.  79.   8x?/-7&2. 
78.   X  ==  2,  y  ~  3.                                80.   G  x  -  23. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED   FOR   FAILURE  TO   RETURN 
THIS    BOOK  ON   THE   DATE  DUE.   THE   PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY    AND    TO    $1.00    ON    THE    SEVENTH     DAY 
OVERDUE. 

NOV  PO  1942 

DEC  2?  .m 

^. 

JM    ^      ... 

^^"^^OBf^ 

REC'D 

Jill 

JUL  1 0  1960 

LD  21-100m-7,'40{6B36s) 

M  1611 


THE  UNIVERSITY  OF  CAUFORNIA  UBRARY 


